#6986494

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 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)(4,6)(5,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(26,31)(27,34)(28,30)(32,39)(35,36)(38,40);(1,3,4,8)(2,5,6,7)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,26,29,31)(27,32,36,38)(28,37,30,33)(34,40,35,39);(3,8)(5,7)(9,10)(11,15)(12,21)(13,16)(14,17)(18,20)(19,24)(22,23)(25,27,29,36)(26,32,31,38)(28,34,30,35)(33,39,37,40);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,30)(26,33,31,37)(27,35,36,34)(32,40,38,39);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(26,31)(32,38)(33,37)(39,40)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6893690);