#72340253

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(4,12)(6,9)(7,14)(13,15)(17,18)(19,24)(20,31)(21,35)(22,26)(23,27)(25,34)(28,38)(29,40)(30,41)(32,46)(33,42)(36,39)(37,43)(44,48)(45,47);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,13,16,15)(17,19)(18,24)(20,32)(21,36)(22,29)(23,30)(25,38)(26,39)(27,41)(28,34)(31,46)(33,45)(35,40)(37,44)(42,48)(43,47);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,20)(18,25,27,31)(19,28,30,32)(21,37)(22,33)(23,34)(24,38)(26,43,35,42)(29,45,36,44)(39,48)(40,47)(41,46);(4,12)(10,16)(17,21)(18,26)(19,29)(20,33)(22,23)(24,39)(25,42)(27,35)(28,44)(30,36)(31,43)(32,45)(34,37)(38,47)(40,41)(46,48);(1,4)(2,7,8,14)(3,10)(5,12)(6,13,9,15)(11,16)(17,22,23,21)(18,26)(19,29)(20,33,34,37)(24,40,41,39)(25,43)(27,35)(28,45)(30,36)(31,42)(32,44)(38,47,46,48)

Polycyclic presentation:

2^10|b^2=j,c^2=gh,e^2=ghij,b^a=bh,c^a=cf,c^b=cfg,d^a=dij,d^b=dfghij,d^c=dij,e^a=ehi,e^b=efgi,e^d=eghi,f^b=fi,f^c=fij,f^e=fj,g^a=gij,g^b=gi,g^c=gj,g^d=gij,g^e=gij,h^c=hj,h^d=hij,h^e=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340258, #72340288, #72340301, …
maximal normal subgroup of	#72340258, #72340288, #72340301, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);