#72310844

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2826 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)(10,16)(17,18)(19,27)(20,24)(21,33)(22,26)(23,30)(25,36)(28,37)(29,38)(31,43)(32,41)(34,42)(35,40)(39,45)(44,48)(46,47);(1,2)(3,9,11,6)(4,7,12,14)(5,8)(10,15)(13,16)(17,19)(18,23,26,27)(20,28,32,45)(21,34)(22,30)(24,39)(25,42,33,38)(29,36)(31,44,35,46)(37,41)(40,48)(43,47);(1,2)(3,6)(4,7,12,14)(5,8)(9,11)(10,15,16,13)(17,20)(18,24)(19,28,30,45)(21,35)(22,32)(23,37,27,39)(25,43)(26,41)(29,44,34,46)(31,36)(33,40)(38,47,42,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21)(18,25)(19,29)(20,31)(22,36)(23,38)(24,40)(26,33)(27,42)(28,44)(30,34)(32,35)(37,47)(39,48)(41,43)(45,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(20,32)(21,36)(25,33)(29,34)(37,39)(38,42)(40,43)(44,46)

Polycyclic presentation:

2^10|b^2=f,c^2=g,b^a=bg,c^a=cg,c^b=cfg,d^a=di,d^b=dh,d^c=dgi,e^a=ehij,e^b=ehij,e^c=efghj,e^d=ei,f^a=fi,f^c=fi,f^d=fj,f^e=fj,g^b=gi,g^e=gj,h^b=hj,h^c=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310874, #72340288, #72340309, …
maximal normal subgroup of	#72310874, #72340288, #72340309, …

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