Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,10)(11,12)(13,14,18,21,20,17,19,27)(15,16,22,25,24,26,23,28);(2,3)(5,6,9,10)(7,11,8,12)(13,15)(14,16)(17,26)(18,22)(19,23)(20,24)(21,25)(27,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,24)(18,28)(19,25)(20,26)(21,23)(22,27);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,17,20,14)(15,16,24,26)(18,27,19,21)(22,25,23,28)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=e,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348273);