#6574403

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	C₂×#3277	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2602 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)(9,10,15,20)(11,16,22,30)(12,25,27,34)(13,19,14,18)(17,28,33,26)(21,35,36,40)(23,32,24,31)(29,38,39,37);(1,2,3,4)(5,6)(7,8)(9,11,15,22)(10,16,20,30)(12,21,27,36)(13,23,14,24)(17,29,33,39)(18,31,19,32)(25,35,34,40)(26,37,28,38);(6,8)(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,29)(18,25)(19,34)(20,33)(22,36)(23,37)(24,38)(30,39)(31,35)(32,40);(5,6)(7,8)(11,22)(16,30)(21,36)(23,24)(29,39)(31,32)(35,40)(37,38);(6,8)(9,13,15,14)(10,18,20,19)(11,23,22,24)(12,26,27,28)(16,31,30,32)(17,25,33,34)(21,37,36,38)(29,35,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^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, 6481599);