Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14175 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 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)(3,13,17,8,16,21,14,23)(4,9,18,24,20,25,19,26)(15,29,31,32,30,27,28,22);(3,14,16,17)(8,21,23,13)(15,28,30,31)(22,29,32,27);(1,3)(2,8)(4,15)(5,17)(6,14)(7,16)(9,22)(10,21)(11,13)(12,23)(18,28)(19,31)(20,30)(24,27)(25,32)(26,29);(1,4,6,18,7,20,5,19)(2,9,11,24,12,25,10,26)(3,15,14,28,16,30,17,31)(8,22,13,27,23,32,21,29)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318864, #72318900, #72318991, … |
| maximal normal subgroup of | #72318864, #72318900, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283368);