#72287716

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22858
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#489284	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=efhj,d^2=eghi,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bj,c^b=cf,d^a=dg,d^b=dfg,d^c=dfg,e^c=ej,e^d=ej,f^b=fh,f^c=fj,f^d=fh,g^a=gij,g^b=gij,g^c=gij,h^b=hj,h^d=hj

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287720, #72320455, #72320616, …
maximal normal subgroup of	#72287720, #72320455, #72320616, …

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