Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,25)(13,20)(14,19)(15,18)(16,23)(17,28)(22,32)(24,35)(26,33)(27,34)(29,31)(30,37)(36,38)(39,40);(1,3,4,7)(2,5,6,8)(9,11,14,29,15,24,13,23)(10,16,20,35,18,31,19,21)(12,22,28,39,26,37,27,38)(17,32,25,36,34,30,33,40);(9,12,13,27,15,26,14,28)(10,17,19,33,18,34,20,25)(11,22,23,38,24,37,29,39)(16,30,21,36,31,32,35,40);(2,6)(5,8)(10,18)(12,26)(16,31)(19,20)(21,35)(22,37)(27,28)(38,39)Polycyclic presentation:
2^9|b^2=eh,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312651);