Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11,16,25)(8,18)(9,21)(10,13)(12,20)(14,22)(15,27,28,36)(17,31,30,24)(19,32)(23,34,35,29)(26,33);(5,7)(6,11,13,25)(8,15)(9,17)(10,16)(12,23,26,35)(14,24,21,31)(18,27,32,36)(19,28)(20,29)(22,30)(33,34);(5,8,9,20,10,19,22,33)(6,12,14,18,13,26,21,32)(7,15,17,29,16,28,30,34)(11,23,24,27,25,35,31,36);(2,4)(8,19)(12,26)(15,28)(18,32)(20,33)(23,35)(27,36)(29,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300209);