#6599447

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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)(11,16)(12,14)(13,15)(17,18,24,21)(19,22,26,23)(20,27,30,32)(25,31,28,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,24,26)(18,22,21,23)(20,27,30,32)(25,29,28,31);(11,12)(14,16)(17,20)(18,25)(19,27)(21,28)(22,29)(23,31)(24,30)(26,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,21,24,18)(19,22,26,23)(20,28,30,25)(27,29,32,31);(11,12)(14,16)(17,22,24,23)(18,26,21,19)(20,29,30,31)(25,32,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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=ehi

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, 6506643);