Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,16)(12,14)(13,15)(17,18,21,26,22,25,20,24)(19,27,29,23,30,32,28,31);(2,4)(5,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,25,31)(20,29,21,28)(24,32,26,27);(2,4)(6,8)(9,12,13,11)(10,14,15,16)(18,24)(19,28)(20,21)(25,26)(27,32)(29,30);(18,25)(23,31)(24,26)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=efi,c^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313037, #72313051, #72313054, … |
| maximal normal subgroup of | #72313037, #72313051, #72313054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265799);