Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,8)(9,10,12,11)(13,14,19,21,18,17,16,22)(15,23,24,20,26,28,25,27);(1,2)(3,4)(13,15,18,26)(14,20,17,27)(16,24,19,25)(21,23,22,28);(2,4)(6,8)(9,11,12,10)(13,16,18,19)(14,21,17,22)(15,24,26,25)(20,23,27,28);(6,8)(13,17,18,14)(15,23,26,28)(16,22,19,21)(20,24,27,25)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efh,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296349);