Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3420 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)(5,6,10,14,11,15,9,22)(7,16,18,12,19,27,23,26)(8,13,20,29,21,30,24,28)(17,32,35,36,34,31,33,25);(5,7,10,18,11,19,9,23)(6,12,14,27,15,26,22,16)(8,17,20,35,21,34,24,33)(13,25,29,32,30,36,28,31);(5,8)(6,13)(7,17)(9,24)(10,20)(11,21)(12,25)(14,29)(15,30)(16,31)(18,35)(19,34)(22,28)(23,33)(26,36)(27,32);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257091, #72257145, #72257147 |
| maximal normal subgroup of | #72257091, #72257145, #72257147 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299775);