#6543676

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 6450872);