#6535570

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(17,18,23,27)(19,25,21,30)(20,26,22,31)(24,28,32,29);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,21)(18,24,27,32)(20,29,22,28)(25,26,30,31);(10,13)(14,16)(17,20)(19,28)(21,29)(22,23)(24,32)(26,31);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(17,21,23,19)(18,25,27,30)(20,29,22,28)(24,26,32,31);(9,11,12,15)(10,14,13,16)(19,21)(24,32)(25,30)(28,29)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442766);