#8118659

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

4506 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)(3,4)(5,6)(7,8)(12,19)(16,21)(17,23)(22,24)(26,29)(27,28)(30,38)(39,40);(2,6)(3,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,26,31,29)(27,32,28,35)(30,33,38,37)(34,39,36,40);(1,3,5,7)(2,4,6,8)(10,14)(15,20)(16,21)(22,24)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(1,4)(2,7)(3,6)(5,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,30,31,38)(26,33,29,37)(27,34,28,36)(32,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8025855);