#72314308

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623305	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,c^2=f,d^2=hj,e^2=fgij,g^2=i,i^2=j,b^a=bh,c^b=ch,d^a=dhj,d^c=dj,e^a=eg,e^b=ej,e^c=eij,e^d=eghi,f^e=fj,g^a=gi,g^c=gj,g^d=gij,g^e=gj,i^a=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314329, #72314408, #72314574
maximal normal subgroup of	#72314329, #72314408, #72314574

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