#72340233

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

2916 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=ghi,d^2=gh,b^a=bf,c^a=cfh,c^b=cfg,d^a=dhi,d^b=dhi,d^c=dghi,e^b=ei,e^c=ehi,e^d=efj,f^c=fi,f^d=fj,g^a=gij,g^b=gi,g^c=gj,g^d=gij,g^e=gj,h^b=hj,h^c=hj,h^d=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340239, #72340248, #72340295, …
maximal normal subgroup of	#72340239, #72340248, #72340295, …

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.6);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, 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.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);