#72309768

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21690
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#491517	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fj,c^2=e,d^2=gj,f^2=h,g^2=i,i^2=j,b^a=bf,d^a=dg,d^b=dhj,d^c=di,e^d=ej,f^a=fh,g^a=gi,g^c=gj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309772, #72310043
maximal normal subgroup of	#72309772, #72310043

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