#6718043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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 ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(9,10,13,15)(11,14,12,16)(19,28)(21,30)(24,32)(26,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,22,27)(19,29,28,26)(20,25,23,31)(21,24,30,32);(10,15)(14,16)(17,19)(18,24)(20,21)(22,28)(23,30)(25,26)(27,32)(29,31);(2,6)(5,7)(17,20,22,23)(18,25,27,31)(19,21,28,30)(24,26,32,29);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(17,21,22,30)(18,26,27,29)(19,20,28,23)(24,25,32,31)

Polycyclic presentation:

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

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, 6625239);