#6573780

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

4639 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)(11,20,23,30)(12,17,24,32)(13,18,14,27)(16,28,31,22)(21,34,35,40)(25,33,26,37)(29,38,39,36);(5,6)(7,8)(9,11,15,23)(10,16,19,31)(12,21,24,35)(13,22,14,28)(17,29,32,39)(18,30,27,20)(25,36,26,38)(33,40,37,34);(2,4)(6,8)(9,12,15,24)(10,17,19,32)(11,21,23,35)(13,25,14,26)(16,29,31,39)(18,33,27,37)(20,34,30,40)(22,36,28,38);(5,6)(7,8)(12,24)(17,32)(21,35)(25,26)(29,39)(33,37)(34,40)(36,38);(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,24,26)(16,30,31,20)(17,33,32,37)(21,36,35,38)(29,40,39,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 6480976);