#72341418

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	#6573661	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#546 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=j,g^2=j,b^a=bfj,c^a=cg,d^a=dfh,d^b=dfj,d^c=dgj,e^b=ehj,e^d=ej,f^c=fi,g^a=gij,g^b=gij,g^c=gj,g^d=gj,g^f=gj,i^a=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341423
maximal normal subgroup of	#72341423

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