#6670209

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

1971 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);(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,24,38)(16,27,29,36)(17,30,33,21)(19,31,34,26)(20,28,37,32)(25,41,40,35)(39,42,44,43);(2,4)(13,15,18,24)(14,19,22,34)(16,25,29,40)(17,31,33,26)(20,35,37,41)(21,38,30,23)(27,44,36,39)(28,43,32,42);(5,7,8,11)(6,10,9,12)(15,24)(19,34)(23,38)(25,40)(26,31)(35,41)(39,44)(42,43);(6,9)(10,12)(13,16,18,29)(14,20,22,37)(15,25,24,40)(17,32,33,28)(19,35,34,41)(21,27,30,36)(23,39,38,44)(26,43,31,42)

Polycyclic presentation:

2^9|b^2=i,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=efhi,h^b=hi,d^c=di,h^c=hi,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, 6577405);