#6908042

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2062 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,26,27,32)(20,25,22,30)(24,28,31,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(18,23)(19,27)(25,30)(28,29);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,19,22,29,21,27,20,28)(18,24,25,26,23,31,30,32);(2,6)(5,7)(10,14)(13,15)(17,18)(19,24)(20,30)(21,23)(22,25)(26,29)(27,31)(28,32);(3,8)(5,7)(17,20,21,22)(18,25,23,30)(19,28,27,29)(24,26,31,32)

Polycyclic presentation:

2^9|a^2=i,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=efh,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, 6815238);