Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1260 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)(5,6,8,10)(7,11,12,9)(13,14,17,22,18,23,19,24)(15,25,26,16,27,20,21,28);(2,3)(5,7,8,12)(6,9,10,11)(13,14)(15,16)(17,24)(18,23)(19,22)(20,21)(25,26)(27,28);(6,10)(9,11)(13,15,17,26,18,27,19,21)(14,20,22,28,23,25,24,16);(1,3,4,2)(7,12)(9,11)(13,16,19,25,18,28,17,20)(14,21,24,27,23,26,22,15)Polycyclic presentation:
2^9|a^2=e,b^2=fi,c^2=gi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351549);