#6671715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2184 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)(13,14,19,18)(15,20,25,32)(16,21,28,33)(17,22,31,30)(23,34,38,26)(24,29,36,39)(27,35,43,42)(37,44,41,40);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,26)(17,29)(18,32)(19,25)(21,23)(22,36)(24,30)(27,41)(28,34)(31,39)(33,38)(35,40)(37,43)(42,44);(2,4)(13,16,19,28)(14,21,18,33)(15,23,25,38)(17,27,31,43)(20,34,32,26)(22,35,30,42)(24,37,36,41)(29,44,39,40);(5,7,8,11)(6,10,9,12)(13,14,19,18)(15,20,25,32)(16,21,28,33)(17,30,31,22)(23,34,38,26)(24,39,36,29)(27,42,43,35)(37,40,41,44);(6,9)(10,12)(13,17,19,31)(14,22,18,30)(15,24,25,36)(16,27,28,43)(20,29,32,39)(21,35,33,42)(23,37,38,41)(26,40,34,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

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, 6578911);