Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(9,10,15,12)(11,18,20,16)(13,22,14,17)(19,23,21,24)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(2,6)(3,4)(9,11)(10,16)(12,18)(13,19)(14,21)(15,20)(17,23)(22,24)(25,26)(27,30)(28,36)(29,32)(31,38)(33,35)(34,40)(37,39);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,20,16)(13,17,14,22)(19,23,21,24)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,4,5,3)(2,8,6,7)(9,13,15,14)(10,17,12,22)(11,19,20,21)(16,23,18,24)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40);(2,6)(7,8)(11,20)(16,18)(19,21)(23,24)(26,32)(30,33)(31,36)(39,40);(9,14,15,13)(10,17,12,22)(11,21,20,19)(16,23,18,24)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007527);