#6461889

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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)(17,18,22,28,23,27,21,26)(19,25,29,32,20,30,31,24);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,27,32)(21,29,22,31)(25,28,30,26);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,25,27,30)(21,29,22,31)(24,28,32,26);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,26,27,28)(19,29,20,31)(24,25,32,30);(18,27)(24,32)(25,30)(26,28)

Polycyclic presentation:

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

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, 6369085);