#6732986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1202 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,31,30)(27,32,37,40)(28,33,36,39)(29,38,35,34);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,38)(29,39)(30,40)(31,37)(33,35)(34,36);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,31,36)(26,33,30,39)(27,35,37,29)(32,34,40,38);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,26,31,30)(27,32,37,40)(28,39,36,33)(29,34,35,38);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(21,23)(25,29)(26,34)(27,36)(28,37)(30,38)(31,35)(32,33)(39,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,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, 6640182);