#6908554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,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, 6815750);