#6484558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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

1608 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)(11,15)(12,16)(13,14)(17,18,22,25,23,20,21,26)(19,27,29,24,30,32,28,31);(2,5)(6,8)(10,14)(11,12)(17,19,23,30)(18,24,20,31)(21,29,22,28)(25,27,26,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,24)(21,26)(22,25)(27,29)(28,32)(30,31);(9,12,13,11)(10,16,14,15);(10,14)(15,16)(17,21,23,22)(18,25,20,26)(19,28,30,29)(24,32,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391754);