Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,8)(10,13)(14,15)(17,18,21,24)(19,20,23,22);(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24);(1,2,4,3)(5,6)(7,8)(17,19)(18,20)(21,23)(22,24);(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23);(1,3,4,2)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^9|a^2=h,c^2=fgi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232234, #72232250, #72232265, … |
| maximal normal subgroup of | #72232234, #72232250, #72232265, … |
| maximal quotient of | #72298262, #72323507, #72323536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1293793);