#6574261

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481457);