#6704354

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1315 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6611550);