#72285509

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	#251827	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2758 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=gj,b^a=bfh,c^a=ch,c^b=cfg,d^a=dfij,d^b=dfh,d^c=dfhij,e^a=efghi,e^b=efgi,e^c=efghi,f^b=fi,f^c=fi,f^e=fj,g^a=gij,g^b=gi,g^d=gj,g^e=gj,h^a=hi,h^b=hi,h^c=hij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599
maximal normal subgroup of	#72285599

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