Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3478 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)(5,6,10,14,11,15,9,24)(7,16,19,27,20,31,18,12)(8,21,22,28,23,13,25,29)(17,32,33,36,34,30,35,26);(5,7,9,18,11,20,10,19)(6,12,14,16,15,27,24,31)(8,17,22,33,23,34,25,35)(13,26,28,30,21,36,29,32);(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,28)(15,21)(16,30)(18,33)(19,35)(20,34)(24,29)(27,36)(31,32);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72249950, #72249958, #72249972, … |
| maximal normal subgroup of | #72249950, #72249958, #72249972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299384);