#6535276

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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)(10,13)(14,16)(17,18,23,21)(19,24,22,31)(20,27,30,32)(25,28,29,26);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,21,31)(20,28,30,26)(25,27,29,32);(10,13)(14,16)(17,20,23,30)(18,25,21,29)(19,26,22,28)(24,32,31,27);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(17,21,23,18)(19,24,22,31)(20,29,30,25)(26,32,28,27);(9,11,12,15)(10,14,13,16)(18,21)(24,31)(25,29)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442472);