#6463690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

1763 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,24,30,23,19,22,27)(20,28,21,26,31,25,32,29);(2,5)(6,7)(10,14)(11,12)(17,19,23,18)(20,28,31,25)(21,29,32,26)(22,27,24,30);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,23,31)(18,25,19,28)(21,22,32,24)(26,30,29,27);(9,12,13,11)(10,15,14,16)(17,21)(18,26)(19,29)(20,22)(23,32)(24,31)(25,30)(27,28);(20,31)(21,32)(25,28)(26,29)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6370886);