#6533690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#23390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(17,18)(19,25)(20,31)(21,27)(22,28)(23,32)(24,29)(26,30);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,23)(21,22)(24,30)(26,29)(27,28)(31,32);(10,13)(14,16)(17,20,24,21)(18,26,29,25)(19,22,30,23)(27,32,31,28);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21,24,20)(18,27,29,31)(19,22,30,23)(25,28,26,32);(11,15)(14,16)(17,22,24,23)(18,28,29,32)(19,20,30,21)(25,31,26,27)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440886);