Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,12,11)(13,14,19,22,18,15,17,21)(16,24,28,20,27,25,26,23);(5,6)(7,8)(13,15,18,14)(16,25,27,24)(17,21,19,22)(20,26,23,28);(2,4)(6,8)(9,11,12,10)(14,15)(20,23)(21,22)(24,25);(2,4)(13,16,18,27)(14,20,15,23)(17,28,19,26)(21,25,22,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,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, 295540);