Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363904 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21379 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,26,31,8,11,25,30,5,10,15,28)(3,14,19,23,17,29,32,4,18,21,24,20,16,9,27,22);(1,3,15,32,5,16,25,19,8,18,26,27,7,17,6,24)(2,9,28,23,10,21,30,22,11,29,31,20,13,14,12,4);(1,4)(2,3)(5,23)(6,14)(7,20)(8,22)(9,15)(10,16)(11,18)(12,24)(13,17)(19,30)(21,25)(26,29)(27,31)(28,32);(2,10,11,13)(3,16,18,17)(6,25)(9,29)(12,31,30,28)(14,21)(15,26)(19,32,24,27)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231607, #72231624, … |
| maximal normal subgroup of | #72213945, #72231607, #72231624, … |
| maximal quotient of | #72224874, #72237201, #72279984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402703);