#6584791

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#37700	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1907 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:

(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,27)(17,21)(18,23)(20,31)(22,28)(24,38)(25,33)(26,34)(29,37)(30,36)(32,41)(35,39)(40,43)(42,44);(1,2)(3,4)(6,10)(11,12)(13,15,18,25)(14,19,23,33)(16,24,29,42)(17,26,28,39)(20,32,36,43)(21,34,22,35)(27,38,37,44)(30,40,31,41);(2,4)(5,7,9,8)(6,11,10,12)(13,16,28,30,18,29,17,31)(14,20,21,37,23,36,22,27)(15,24,39,40,25,42,26,41)(19,32,34,44,33,43,35,38);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,25)(16,29)(19,33)(20,36)(26,39)(27,37)(30,31)(34,35);(2,4)(13,17,18,28)(14,21,23,22)(15,26,25,39)(16,30,29,31)(19,34,33,35)(20,27,36,37)(24,40,42,41)(32,38,43,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491987);