Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188770 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#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×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1021 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)(7,12,19,30)(8,21)(9,14,24,32)(10,16)(11,26)(13,23)(15,17)(18,28)(20,35,33,29)(22,36)(25,27,34,31);(1,2)(3,4)(5,7,17,33,10,19,11,20)(6,12,15,29,16,30,26,35)(8,9,18,34,22,24,23,25)(13,27,21,14,28,31,36,32);(2,4)(5,8,11,23,10,22,17,18)(6,13,15,21,16,28,26,36)(7,9,20,25,19,24,33,34)(12,27,29,14,30,31,35,32);(5,9)(6,14)(7,18)(8,20)(10,24)(11,25)(12,28)(13,30)(15,31)(16,32)(17,34)(19,23)(21,35)(22,33)(26,27)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268098);