Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14175 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,10)(3,13,16,25,17,24,15,8)(4,18,21,28,22,27,20,9)(14,29,19,32,31,23,30,26);(1,3,5,15,7,17,6,16)(2,8,11,13,12,25,10,24)(4,19,20,14,22,30,21,31)(9,26,18,29,28,32,27,23);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,27)(11,18)(12,28)(13,26)(15,30)(16,19)(17,31)(24,32)(25,29);(2,10,12,11)(8,24,25,13)(9,27,28,18)(23,32,29,26)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215981, #72216141, #72219684, … |
| maximal normal subgroup of | #72215981, #72216141, #72219684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283387);