#6461807

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369003);