Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,27)(7,18,20,13,21,15,9,26)(8,22,24,31,25,14,28,32)(19,30,23,29,35,33,34,36);(1,2)(3,4)(5,7)(6,13)(8,19)(9,10)(11,20)(12,21)(14,29)(15,16)(17,26)(18,27)(22,36)(23,24)(25,35)(28,34)(30,31)(32,33);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,28)(12,25)(13,29)(15,30)(16,31)(17,22)(18,33)(20,34)(21,35)(26,36)(27,32);(5,9,11,21,12,20,10,7)(6,15,27,26,17,18,16,13)(8,23,28,35,25,34,24,19)(14,30,32,36,22,33,31,29)Polycyclic presentation:
2^9|a^2=h,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290743);