#6810295

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2023 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)(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,25,31)(16,20,27,33)(17,21,28,34)(23,29,36,40)(24,30,37,41)(26,32,38,42)(35,39,43,44);(5,7,8,12)(6,9,10,11)(15,23)(16,26)(17,28)(19,29)(20,32)(21,34)(24,37)(25,36)(27,38)(30,41)(31,40)(33,42);(2,4)(7,12)(9,11)(13,15,17,23,18,25,28,36)(14,19,21,29,22,31,34,40)(16,24,26,35,27,37,38,43)(20,30,32,39,33,41,42,44);(6,10)(9,11)(16,27)(20,33)(24,37)(26,38)(30,41)(32,42)(35,43)(39,44);(13,16,17,26,18,27,28,38)(14,20,21,32,22,33,34,42)(15,24,23,35,25,37,36,43)(19,30,29,39,31,41,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6717491);