Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,30)(21,32)(22,24)(23,29)(25,28)(26,31);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,23,24,30)(20,31,29,21)(25,32,27,26);(2,4)(10,14)(13,15)(18,24)(19,28)(20,29)(26,32);(11,16)(13,15)(17,20,21,19,22,29,31,28)(18,25,26,23,24,27,32,30)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292046);