#6584765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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

1909 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)(16,28)(17,22)(18,32)(19,23)(21,31)(24,37)(25,34)(26,35)(27,41)(29,42)(30,36)(33,39)(38,43)(40,44);(1,2)(3,4)(6,10)(11,12)(13,15,19,25)(14,20,23,34)(16,24,29,40)(17,26,18,27)(21,33,36,43)(22,35,32,41)(28,37,42,44)(30,38,31,39);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,31,19,29,17,30)(14,21,22,28,23,36,32,42)(15,24,27,39,25,40,26,38)(20,33,35,37,34,43,41,44);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,25)(16,29)(20,34)(21,36)(26,27)(28,42)(30,31)(35,41);(2,4)(13,17,19,18)(14,22,23,32)(15,26,25,27)(16,30,29,31)(20,35,34,41)(21,28,36,42)(24,38,40,39)(33,37,43,44)

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,d^b=di,e^b=eh,d^c=dhi,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, 6491961);