#72286529

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	C₄×C₈	C₂⁵
prime radical	#7624074	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17320 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=ghi,d^2=f,f^2=h,g^2=i,i^2=j,b^a=bhj,d^b=dghi,d^c=dg,e^a=ehj,e^b=eg,e^c=eghi,e^d=eghj,f^b=fj,f^c=fj,f^e=fj,g^d=gi,g^e=gij,i^d=ij,i^e=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286625, #72287030, #72287214, …
maximal normal subgroup of	#72286625, #72287030, #72287214, …

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