#72306803

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13584
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	#493	C₂⁴
prime radical	#6357800	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ #60639 ◁ G
LECS	G ▷ #493 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=e,c^2=gi,g^2=ij,h^2=j,b^a=bij,c^a=cj,c^b=cef,d^b=di,d^c=dij,e^a=ej,e^c=eg,e^d=ej,f^a=fj,f^b=fghi,f^c=fgj,f^e=fhj,g^b=gh,g^e=gij,g^f=gij,h^b=hi,h^e=hj,h^f=hj,i^b=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306806, #72306826, #72306979, …
maximal normal subgroup of	#72306806, #72306826, #72306979, …

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