Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,12,11)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,28)(25,27);(5,6)(7,8)(13,15,17,23)(14,18,19,27)(16,25,26,22)(20,28,24,21);(2,4)(6,8);(9,11,12,10)(14,19)(18,27)(20,24)(21,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=cg,e^b=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264369, #72264372, #72264442, … |
| maximal normal subgroup of | #72264369, #72264372, #72264442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266199);