#6469053

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1836 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,24)(19,25,27,31)(20,26,21,30)(23,29,32,28);(9,10)(11,16)(12,14)(13,15)(17,19,22,27)(18,23,24,32)(20,28,21,29)(25,26,31,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(18,24)(19,27)(26,30)(28,29);(10,14)(13,16)(17,19)(18,25)(20,29)(21,28)(22,27)(23,30)(24,31)(26,32);(1,4,5,3)(2,6,7,8)(17,20,22,21)(18,26,24,30)(19,28,27,29)(23,31,32,25)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6376249);