#6612093

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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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)(9,10)(11,14)(12,16)(13,15)(17,18,23,24,22,28,21,27)(19,29,31,32,30,25,20,26);(9,11,13,12)(10,14,15,16)(18,24)(19,30)(21,23)(25,32)(26,29)(27,28);(2,5)(6,8)(10,15)(11,12)(17,19)(18,25)(20,21)(22,30)(23,31)(24,26)(27,32)(28,29);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(17,20,22,31)(18,26,28,32)(19,21,30,23)(24,29,27,25);(11,12)(14,16)(17,21,22,23)(18,27,28,24)(19,31,30,20)(25,26,29,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6519289);