#72265093

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	#7623715	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ D₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3099 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)(3,6)(4,8)(5,7);(1,3)(2,6)(4,5)(7,8)(9,10)(11,21,24,36)(12,26,27,37)(13,29)(14,31)(15,20)(16,25,35,39)(17,28,33,40)(18,30)(19,32)(22,38)(23,34);(11,12,24,27)(13,22)(16,33,35,17)(18,34)(21,26,36,37)(23,30)(25,40,39,28)(29,38);(1,4,5,3)(2,6,7,8)(9,11,22,39,14,12,23,40,15,24,13,25,32,27,30,28)(10,16,34,36,19,33,38,26,20,35,18,21,31,17,29,37);(3,4)(6,8)(9,12,13,28,14,24,30,39,15,27,22,40,32,11,23,25)(10,17,18,26,19,16,29,21,20,33,34,37,31,35,38,36)

Polycyclic presentation:

2^10|b^2=f,c^2=f,d^2=gj,e^2=gh,g^2=i,i^2=j,d^a=dh,d^b=dg,d^c=di,e^b=eg,e^c=ei,e^d=eh,f^d=fj,f^e=fj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265164, #72265340, #72265527, …
maximal normal subgroup of	#72265164, #72265340, #72265527, …

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