#6624625

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6531821);