Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
839 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)(7,10)(8,11)(9,12)(13,14,16,20,18,21,17,26)(15,22,24,19,25,23,27,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,23)(16,26)(17,20)(18,21)(19,24)(22,25)(27,28);(2,4)(5,7,9,8)(6,11,12,10);(5,8,9,7)(6,11,12,10)(13,15,16,24,18,25,17,27)(14,19,20,23,21,28,26,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=f,d^2=ef,e^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,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339038);