#6468232

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1646 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)(9,10)(11,15)(12,16)(13,14)(17,18,23,26,22,19,20,24)(21,25,29,32,30,27,28,31);(10,14)(11,12)(17,19,22,18)(20,24,23,26)(21,25,30,27)(28,32,29,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,24,19,26)(21,29,30,28)(25,32,27,31);(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,25,19,27)(20,28,23,29)(24,31,26,32);(1,3,4,7)(2,6,5,8)(21,30)(25,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,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, 6375428);