#6461983

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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

1540 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)(4,7)(5,6)(17,18,23,25,22,28,20,27)(19,29,31,24,21,26,30,32);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,22)(23,30)(25,29)(26,27)(28,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,22,23)(18,25,28,27)(19,30,21,31)(24,26,32,29);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,19)(18,26,28,29)(20,31,23,30)(24,25,32,27);(17,20,22,23)(18,25,28,27)(19,31,21,30)(24,29,32,26)

Polycyclic presentation:

2^9|a^2=fg,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6369179);