#6599413

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

1820 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)(17,18,22,24)(19,28,20,26)(21,27,23,31)(25,30,32,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,24)(19,20)(27,31)(29,30);(11,12)(14,16)(17,19)(18,25)(20,22)(21,29)(23,30)(24,32)(26,27)(28,31);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,20,22,19)(18,26,24,28)(21,29,23,30)(25,31,32,27);(11,12)(14,16)(17,21,22,23)(18,27,24,31)(19,29,20,30)(25,26,32,28)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=f,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, 6506609);