Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,23)(20,24);(5,6)(7,8)(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24);(1,2,3,4)(6,8);(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,18,23,24);(11,20)(15,23)(18,24)(19,21)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72282243, #72285048, #72313050, … |
| maximal normal subgroup of | #72282243, #72285048, #72313050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350294);