#6574401

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2614 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)(11,16,24,32)(12,25,27,33)(13,18,14,19)(17,28,34,26)(21,35,38,40)(22,30,23,31)(29,36,39,37);(1,2,3,4)(5,6)(7,8)(9,11)(10,16)(12,21)(13,23)(14,22)(15,24)(17,29)(18,31)(19,30)(20,32)(25,35)(26,37)(27,38)(28,36)(33,40)(34,39);(6,8)(9,12,15,27)(10,17,20,34)(11,21,24,38)(13,26,14,28)(16,29,32,39)(18,33,19,25)(22,36,23,37)(30,35,31,40);(5,6)(7,8)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,26,27,28)(16,30,32,31)(17,33,34,25)(21,36,38,37)(29,35,39,40);(6,8)(9,13,15,14)(10,19,20,18)(11,23,24,22)(12,26,27,28)(16,30,32,31)(17,25,34,33)(21,37,38,36)(29,35,39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,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, 6481597);