#6816196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,21)(15,23,26,19)(16,20,29,34)(18,22,25,31)(24,35,37,30)(27,36,40,32)(28,33,39,42)(38,43,44,41);(5,7,8,12)(6,9,10,11)(15,24)(16,28)(18,25)(19,30)(20,33)(22,31)(23,35)(26,37)(27,40)(29,39)(32,36)(34,42);(1,2,3,4)(6,10)(7,12)(13,15,25,37,17,26,18,24)(14,19,31,35,21,23,22,30)(16,27,39,44,29,40,28,38)(20,32,42,43,34,36,33,41);(6,10)(9,11)(15,26)(19,23)(24,37)(27,40)(30,35)(32,36)(38,44)(41,43);(13,16,25,39,17,29,18,28)(14,20,31,42,21,34,22,33)(15,27,37,44,26,40,24,38)(19,32,35,43,23,36,30,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723392);