#6810616

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

1816 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)(6,10)(7,8)(13,14,18,22)(15,19,24,30)(16,20,28,34)(17,21,27,33)(23,29,35,39)(25,31,37,41)(26,32,38,42)(36,40,43,44);(5,6)(7,12)(8,11)(9,10)(15,23)(16,26)(17,27)(19,29)(20,32)(21,33)(24,35)(25,37)(28,38)(30,39)(31,41)(34,42);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,23,18,24,27,35)(14,19,21,29,22,30,33,39)(16,25,26,36,28,37,38,43)(20,31,32,40,34,41,42,44);(5,8,9,7)(6,11,10,12)(15,24)(19,30)(23,35)(25,37)(29,39)(31,41)(36,43)(40,44);(13,16,27,38,18,28,17,26)(14,20,33,42,22,34,21,32)(15,25,35,43,24,37,23,36)(19,31,39,44,30,41,29,40)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,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, 6717812);