#8602778

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509974);