Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,22,24,28)(16,20)(17,26)(18,21)(19,25,27,23);(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(15,23,24,25)(19,22,27,28);(6,9)(7,12)(10,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,28)(21,27)(22,26)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,e^b=ei,f^b=fi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23493);