#6671681

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

2191 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)(13,14,18,22)(15,19,25,33)(16,20,27,35)(17,21,28,36)(23,31,29,37)(24,32,40,43)(26,34,30,38)(39,42,41,44);(5,6,8,9)(7,10,11,12)(15,23)(16,26)(17,28)(19,31)(20,34)(21,36)(25,29)(27,30)(33,37)(35,38)(39,41)(42,44);(2,4)(13,15,18,25)(14,19,22,33)(16,24,27,40)(17,29,28,23)(20,32,35,43)(21,37,36,31)(26,39,30,41)(34,42,38,44);(5,7,8,11)(6,10,9,12)(16,27)(20,35)(24,40)(26,30)(32,43)(34,38)(39,41)(42,44);(6,9)(10,12)(13,16,18,27)(14,20,22,35)(15,24,25,40)(17,30,28,26)(19,32,33,43)(21,38,36,34)(23,39,29,41)(31,42,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6578877);