#6807981

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

1857 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)(13,14,18,23)(15,20,26,35)(16,28,31,21)(17,22,27,36)(19,24,30,38)(25,41,43,34)(29,40,33,37)(32,44,42,39);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,20,23,35)(16,29,31,33)(17,30,27,19)(21,37,28,40)(22,38,36,24)(25,42,43,32)(34,44,41,39);(2,4)(13,16,30,42,18,31,19,32)(14,21,38,44,23,28,24,39)(15,25,17,33,26,43,27,29)(20,34,22,40,35,41,36,37);(5,7,8,11)(6,10,9,12)(16,31)(21,28)(25,43)(29,33)(32,42)(34,41)(37,40)(39,44);(6,9)(10,12)(13,17,19,15,18,27,30,26)(14,22,24,20,23,36,38,35)(16,29,32,43,31,33,42,25)(21,37,39,41,28,40,44,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6715177);