#6616268

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(17,18)(19,29)(20,25)(21,26)(22,32)(23,28)(24,30)(27,31);(9,10,12,13)(11,14,15,16)(17,19,23,21)(18,24,28,25)(20,31,30,22)(26,32,29,27);(1,3,5,4)(2,6,8,7)(10,13)(11,15)(17,20)(18,25)(19,22)(21,31)(23,30)(24,28)(26,27)(29,32);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21,23,19)(18,26,28,29)(20,31,30,22)(24,32,25,27);(1,4,5,3)(2,6,8,7)(10,13)(11,15)(19,21)(20,30)(24,25)(26,29)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6523464);