#6810461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

2025 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)(17,30)(22,38)(26,32)(29,33)(31,43)(36,39)(40,44)(41,42);(5,7,8,12)(6,9,10,11)(13,14)(15,24)(16,21)(17,31)(18,34)(19,23)(20,25)(22,33)(26,39)(27,35)(28,37)(29,38)(30,43)(32,36)(40,41)(42,44);(2,4)(7,12)(9,11)(13,15,18,25,19,27,16,28)(14,20,21,24,23,37,34,35)(17,32,33,42,30,26,29,41)(22,39,31,44,38,36,43,40);(6,10)(9,11)(13,16,19,18)(14,21,23,34)(15,25,27,28)(17,29,30,33)(20,35,37,24)(22,31,38,43)(26,41,32,42)(36,44,39,40);(13,17,18,33,19,30,16,29)(14,22,21,31,23,38,34,43)(15,26,25,41,27,32,28,42)(20,36,24,40,37,39,35,44)

Polycyclic presentation:

2^9|b^2=f,c^2=fh,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6717657);