#6461683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1654 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)(17,18)(19,26)(20,24)(21,31)(22,25)(23,28)(27,30)(29,32);(9,10)(11,15)(12,14)(13,16)(17,19,21,28,22,29,20,30)(18,23,24,26,25,27,31,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,31)(21,24)(22,25)(26,28)(27,29)(30,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,18)(19,27)(20,31)(21,24)(22,25)(23,29)(26,30)(28,32);(17,20,22,21)(18,24,25,31)(19,28,29,30)(23,32,27,26)

Polycyclic presentation:

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

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, 6368879);