Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,17,20,23,18,22)(15,21,24,26,25,28,27,16);(1,2,4,3)(6,9)(10,12)(13,15)(14,21)(16,22)(17,26)(18,27)(19,24)(20,25)(23,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,24)(18,28)(19,21)(20,26)(22,27)(23,25);(13,17,19,23,20,22,18,14)(15,16,24,21,25,26,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266447);