Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #360200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2311 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13956 |
| prime radical | #62686 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2543 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)(3,13)(4,17,19,26)(5,10)(6,11)(7,12)(8,15)(9,18,27,21)(14,25,31,22)(16,23)(20,24)(28,30,32,29);(1,3)(2,8,12,24)(4,14)(5,15)(6,20)(7,16)(9,25,27,22)(10,23,11,13)(17,28,26,32)(18,29)(19,31)(21,30);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,22)(10,17)(11,26)(12,27)(13,28)(15,29)(16,31)(20,30)(23,32)(24,25);(2,10,12,11)(8,23,24,13)(9,17,27,26)(22,32,25,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285951, #72286033, #72286186, … |
| maximal normal subgroup of | #72285951, #72286033, #72286186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271119);