#6828590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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 ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1187 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)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,29,31)(27,35,37,32)(28,38,36,33)(30,34,39,40);(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(26,31)(27,36)(28,37)(30,39)(32,33)(35,38);(2,5)(6,8)(25,27,30,36,29,37,39,28)(26,32,34,38,31,35,40,33);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,31)(27,37)(28,36)(34,40);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,28,30,27,29,36,39,37)(26,33,34,32,31,38,40,35)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=fg,e^2=fhi,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,e^d=ei

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, 6735786);