#6616271

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

1561 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)(18,23)(21,29)(24,31)(25,30)(26,32)(27,28);(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,31)(28,32);(1,3,5,4)(2,6,8,7)(10,13)(11,15)(17,19,22,20)(18,24,25,26)(21,27,29,28)(23,31,30,32);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,24,25,26)(21,28,29,27)(23,31,30,32);(1,4,5,3)(2,6,8,7)(10,13)(11,15)(18,25)(19,20)(23,30)(27,28)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,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, 6523467);