#6846830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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)(4,8)(5,6)(9,10,13,15)(11,12,16,14)(17,18,22,26)(19,27,29,23)(20,24,21,25)(28,32,30,31);(9,11,13,16)(10,12,15,14)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,32)(26,27);(2,6)(7,8)(9,11,10,14,13,16,15,12)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32);(1,3,5,4)(2,7,6,8)(9,10,13,15)(11,12,16,14)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,32,27,31);(1,4,5,3)(2,8,6,7)(9,12,10,11,13,14,15,16)(17,21,22,20)(18,25,26,24)(19,28,29,30)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fh,d^2=fg,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6754026);