#6573974

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×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2595 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)(11,16,23,31)(12,24,26,34)(13,19,14,18)(17,28,32,25)(21,35,37,39)(22,33,27,30)(29,36,40,38);(1,2,3,4)(5,6)(7,8)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,33)(19,30)(20,31)(24,35)(25,38)(26,37)(28,36)(32,40)(34,39);(6,8)(9,12,15,26)(10,17,20,32)(11,21,23,37)(13,25,14,28)(16,29,31,40)(18,24,19,34)(22,36,27,38)(30,39,33,35);(5,6)(7,8)(9,13,15,14)(10,18,20,19)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,24,32,34)(21,36,37,38)(29,39,40,35);(6,8)(11,23)(16,31)(21,37)(22,27)(29,40)(30,33)(35,39)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,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, 6481170);