#6826672

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,29)(22,27)(23,25)(26,32)(28,31);(9,11)(10,13)(12,16)(14,15)(17,18,22,27)(19,21,28,32)(20,25,23,30)(24,26,31,29);(2,5)(6,8)(11,16)(13,15)(17,19,20,26,22,28,23,29)(18,24,30,32,27,31,25,21);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,23)(18,25,27,30)(19,26,28,29)(21,31,32,24);(17,21,23,31,22,32,20,24)(18,26,25,19,27,29,30,28)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=gi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,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, 6733868);