#6584787

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	#37701	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1893 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)(13,14,19,23)(15,20,25,34)(16,28,29,37)(17,22,18,32)(21,31,36,30)(24,38,40,43)(26,35,27,42)(33,41,44,39);(1,2)(3,4)(6,10)(11,12)(13,15,19,25)(14,20,23,34)(16,24,29,40)(17,26,18,27)(21,33,36,44)(22,35,32,42)(28,38,37,43)(30,39,31,41);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,30,19,29,17,31)(14,21,22,28,23,36,32,37)(15,24,27,39,25,40,26,41)(20,33,35,38,34,44,42,43);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,25)(16,29)(20,34)(21,36)(26,27)(28,37)(30,31)(35,42);(2,4)(13,17,19,18)(14,22,23,32)(15,26,25,27)(16,30,29,31)(20,35,34,42)(21,37,36,28)(24,39,40,41)(33,43,44,38)

Polycyclic presentation:

2^9|a^2=i,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^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, 6491983);