#10218493

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

4518 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:

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

Polycyclic presentation:

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

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, 10125689);