Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,20)(18,26)(22,25)(23,28)(24,27);(1,2)(3,4)(6,9)(10,12)(14,19)(20,27)(21,26)(22,28);(2,4)(14,20)(15,18)(17,25)(19,27)(21,22)(23,24)(26,28);(5,7,8,11)(6,10,9,12)(13,15,16,18)(14,21,19,26)(17,23,24,25)(20,28,27,22)Polycyclic presentation:
2^9|d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^c=di,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225017, #72225053, #72225064, … |
| maximal normal subgroup of | #72225017, #72225053, #72225064, … |
| maximal quotient of | #72224983, #72224986, #72224988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406431);