Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2283 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,16,19,12)(8,13)(9,14)(10,15)(11,18,24,32)(17,29,31,25)(20,26)(21,27)(22,28)(23,30,34,36)(33,35);(1,2)(3,4)(5,7)(6,11,15,24)(8,17)(9,18)(10,19)(12,14,16,28)(13,23,27,34)(20,30)(21,31)(22,32)(25,26,29,35)(33,36);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,14)(13,25)(15,16)(20,30)(21,31)(22,32)(23,26)(24,28)(27,29)(33,36)(34,35);(5,8,9,20,10,21,22,33)(6,13,14,26,15,27,28,35)(7,17,18,30,19,31,32,36)(11,23,16,29,24,34,12,25)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265614);