#8754887

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

4327 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662083);