Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,13)(9,16)(10,14)(11,15)(12,20)(18,32)(19,26)(21,27)(22,29)(23,31)(24,30)(25,35)(28,34)(33,36);(5,7,10,21,11,19,9,20)(6,12,16,26,15,27,14,17)(8,18,22,34,23,33,24,35)(13,28,30,32,31,25,29,36);(2,4)(5,8)(6,13)(7,18)(9,24)(10,22)(11,23)(12,25)(14,29)(15,31)(16,30)(17,32)(19,33)(20,35)(21,34)(26,36)(27,28);(6,14,15,16)(7,19)(12,26,27,17)(13,29,31,30)(18,33)(20,21)(25,36,28,32)(34,35)Polycyclic presentation:
2^9|b^2=h,d^2=g,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^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231526, #72252977, #72256905, … |
| maximal normal subgroup of | #72231526, #72252977, #72256905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318510);