#6573766

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4644 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)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,35)(23,30)(24,32)(25,33)(26,34)(28,31)(29,37)(36,40)(38,39);(5,6)(7,8)(9,11,15,23)(10,16,19,31)(12,21,26,38)(13,22,14,28)(17,29,34,40)(18,30,27,20)(24,36,25,37)(32,35,33,39);(2,4)(6,8)(9,12,15,26)(10,17,19,34)(11,21,23,38)(13,25,14,24)(16,29,31,40)(18,33,27,32)(20,35,30,39)(22,37,28,36);(5,6)(7,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,26,25)(16,30,31,20)(17,32,34,33)(21,36,38,37)(29,35,40,39);(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,26,24)(16,30,31,20)(17,33,34,32)(21,37,38,36)(29,39,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480962);