#6884122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,23,26)(19,24,27,32)(20,22,29,31)(21,25,30,28);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,28)(21,22)(23,27)(25,29)(26,32)(30,31);(2,6)(3,8)(10,14)(15,16)(17,20,22,26,23,29,31,18)(19,25,21,24,27,28,30,32);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,30)(18,25,26,28)(19,22,27,31)(20,24,29,32);(9,12,13,11)(10,15,14,16)(19,27)(21,30)(24,32)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6791318);