#6808001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1863 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=f,e^2=h,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

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, 6715197);