Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,18,23,19,21,25,22,26)(20,27,30,24,28,32,29,31);(5,6,7,8)(10,13)(14,16)(17,19,23,25,21,26,22,18)(20,24,30,32,28,31,29,27);(2,4)(6,8)(11,15)(14,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32);(9,11,12,15)(10,14,13,16)(20,28)(24,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240763, #72240772, #72240784, … |
| maximal normal subgroup of | #72240763, #72240772, #72240784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287834);