Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,15,11,16,19,14,22)(12,17,23,20,24,18,13,21);(5,6)(7,8)(9,11)(10,14)(12,20)(13,17)(15,19)(16,22)(18,23)(21,24);(2,4)(6,8)(9,12,15,23,16,24,14,13)(10,17,11,20,19,18,22,21);(2,4)(9,13,14,24,16,23,15,12)(10,18,11,21,19,17,22,20)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295713);