#6601447

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	#23972	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(4,8)(5,6)(17,18)(19,28)(20,25)(21,26)(22,31)(23,27)(24,29)(30,32);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(19,20)(24,32)(25,28)(29,30);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,19)(18,24)(20,23)(21,29)(22,30)(25,26)(27,32)(28,31);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(17,20,23,19)(18,25,27,28)(21,29,22,30)(24,31,32,26);(10,13)(11,15)(17,21,23,22)(18,26,27,31)(19,29,20,30)(24,25,32,28)

Polycyclic presentation:

2^9|b^2=hi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,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, 6508643);