#6710178

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	#42584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,32)(27,35,39,33)(28,30,34,29)(36,38,37,40);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,33)(28,38)(29,37)(30,36)(31,39)(32,35)(34,40);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,30)(27,36)(29,32)(31,34)(33,40)(35,38)(37,39);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29)(26,34)(27,37)(28,32)(30,31)(33,40)(35,38)(36,39);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,31,29)(26,34,32,28)(27,38,39,40)(33,36,35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fgi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6617374);