#6810626

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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)(6,10)(7,8);(5,6)(7,12)(8,11)(9,10)(13,14,19,23)(15,24,27,34)(16,21,18,33)(17,30,32,43)(20,28,36,25)(22,29,37,31)(26,40,41,35)(38,42,44,39);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,28,19,27,18,25)(14,20,33,34,23,36,21,24)(17,26,29,42,32,41,31,39)(22,35,43,44,37,40,30,38);(5,8,9,7)(6,11,10,12)(13,16,19,18)(14,21,23,33)(15,25,27,28)(17,29,32,31)(20,34,36,24)(22,30,37,43)(26,39,41,42)(35,44,40,38);(13,17,16,29,19,32,18,31)(14,22,33,43,23,37,21,30)(15,26,28,42,27,41,25,39)(20,35,34,44,36,40,24,38)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717822);