#6996110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1714 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(18,24)(19,26)(21,23)(25,30)(27,28)(29,32);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,23)(21,31)(22,24)(25,29)(26,30)(28,32);(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,30)(21,26)(22,29)(23,32)(24,27)(28,31);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,21)(19,28)(22,31)(23,24)(25,32)(26,27)(29,30);(10,14)(13,15)(17,21,22,23)(18,20,24,31)(19,26,29,32)(25,30,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903306);