#6574406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481602);