#6585004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 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)(15,24)(16,28)(17,22)(18,23)(19,33)(20,26)(21,31)(25,39)(27,36)(29,43)(30,37)(32,35)(34,44)(38,40)(41,42);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,20,23,24)(16,25,29,41)(17,32,19,27)(21,34,37,38)(22,36,33,35)(28,42,43,39)(30,44,31,40);(2,4)(13,16,19,31,18,29,17,30)(14,21,22,28,23,37,33,43)(15,25,27,40,26,41,32,44)(20,34,36,42,24,38,35,39);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(15,26)(16,29)(20,24)(21,37)(27,32)(28,43)(30,31)(35,36);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,19)(14,22,23,33)(15,27,26,32)(16,30,29,31)(20,35,24,36)(21,28,37,43)(25,40,41,44)(34,39,38,42)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492200);