#6698452

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1234 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)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,35)(28,30)(29,34)(31,33)(32,38)(36,39)(37,40);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,38)(26,32,28,35)(29,37,31,39)(33,40,34,36);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,26)(27,36,38,40)(29,34,31,33)(32,39,35,37);(1,3,4,7)(2,6,5,8)(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,33,28,34)(27,37,38,39)(32,40,35,36);(11,19)(14,17)(18,24)(21,23)(25,29)(26,34)(27,38)(28,33)(30,31)(32,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6605648);