#6632418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,30)(21,26)(22,27)(23,32)(24,29)(25,31);(1,3)(2,5)(4,7)(6,8)(10,13)(11,15)(17,19,22,29)(18,24,27,28)(20,21,31,23)(25,26,30,32);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,23)(21,29)(22,31)(25,26)(27,28)(30,32);(3,7)(5,8)(10,13)(14,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,30)(28,32);(2,6)(5,8)(17,21,22,23)(18,26,27,32)(19,20,29,31)(24,25,28,30)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539614);