#6622809

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#25397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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)(9,10)(11,15)(12,16)(13,14)(17,18,22,20)(19,26,27,32)(21,24,28,25)(23,29,31,30);(2,6)(7,8)(10,14)(11,12)(19,27)(23,31)(26,32)(29,30);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19,28,29,22,27,21,30)(18,23,24,32,20,31,25,26);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,20,22,18)(19,23,27,31)(21,24,28,25)(26,29,32,30);(9,12,13,11)(10,16,14,15)(17,21,22,28)(18,24,20,25)(19,29,27,30)(23,26,31,32)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6530005);