#6461634

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

1705 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)(4,5)(7,8)(17,18,22,25)(19,26,24,21)(20,23,28,30)(27,32,31,29);(9,10)(11,15)(12,14)(13,16)(17,19,21,18,22,24,26,25)(20,27,29,23,28,31,32,30);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,30)(26,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15);(20,28)(23,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6368830);