#6584657

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 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)(15,20)(16,27)(17,22)(18,33)(19,23)(21,31)(24,40)(25,35)(26,37)(28,42)(29,43)(30,39)(32,36)(34,41)(38,44);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,20,23,37)(16,28,29,24)(17,32,18,25)(21,38,39,34)(22,36,33,35)(27,42,43,40)(30,41,31,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,31,19,29,17,30)(14,21,22,27,23,39,33,43)(15,24,25,41,26,28,32,44)(20,34,36,40,37,38,35,42);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(16,29)(21,39)(24,28)(27,43)(30,31)(34,38)(40,42)(41,44);(2,4)(13,17,19,18)(14,22,23,33)(15,25,26,32)(16,30,29,31)(20,35,37,36)(21,27,39,43)(24,41,28,44)(34,42,38,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

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, 6491853);