Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,16)(17,25)(18,24)(19,21)(20,22)(23,27)(26,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,19,18)(14,16,21,24)(17,22,27,26)(20,23,28,25);(2,4)(13,16,15,14,19,24,18,21)(17,23,22,20,27,25,26,28);(7,12)(9,11)(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330234);