#6811387

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

2048 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)(7,10)(8,12)(9,11)(17,30)(22,37)(26,40)(29,32)(31,43)(35,41)(38,44)(39,42);(5,7,10,6)(8,12,9,11)(13,14)(15,24)(16,21)(17,31)(18,33)(19,23)(20,25)(22,32)(26,41)(27,34)(28,36)(29,37)(30,43)(35,40)(38,39)(42,44);(2,4)(13,15,18,25,19,27,16,28)(14,20,21,24,23,36,33,34)(17,26,32,39,30,40,29,42)(22,35,31,38,37,41,43,44);(5,8,10,9)(6,11,7,12)(13,16,19,18)(14,21,23,33)(15,25,27,28)(17,29,30,32)(20,34,36,24)(22,31,37,43)(26,39,40,42)(35,44,41,38);(5,9,10,8)(6,11,7,12)(13,17,16,29,19,30,18,32)(14,22,33,43,23,37,21,31)(15,26,28,42,27,40,25,39)(20,35,34,44,36,41,24,38)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=fi,e^2=fhi,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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6718583);