Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1348 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,21)(16,29)(17,20)(18,23)(19,24)(22,32)(25,41)(26,35)(27,34)(28,37)(30,44)(31,40)(33,39)(36,42)(38,43);(5,6,8,9)(7,10,11,12)(14,20,24,23)(21,35,37,34)(22,29,40,39)(36,41,44,38);(2,4)(7,11)(10,12)(13,15,19,28)(14,21,24,37)(16,30,33,42)(17,27,18,26)(20,34,23,35)(22,38,40,41)(25,32,43,31)(29,44,39,36);(5,7,8,11)(6,10,9,12)(13,16,18,32,19,33,17,31)(14,22,20,29,24,40,23,39)(15,25,27,42,28,43,26,30)(21,36,35,41,37,44,34,38)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379206);