Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1142 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)(7,12)(8,21,23,30)(9,24,26,34)(10,15)(11,16)(13,17,35,27)(14,22,29,25)(18,31)(19,32)(20,33)(28,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,17)(10,18)(11,20)(13,34)(14,30)(15,31)(16,33)(19,28)(21,29)(23,25)(24,35)(26,27)(32,36);(5,8,10,27,11,23,28,17)(6,13,15,30,16,35,36,21)(7,9,18,22,20,26,19,25)(12,29,31,34,33,14,32,24);(2,4)(5,9)(6,14)(7,17)(8,18)(10,22)(11,26)(12,30)(13,32)(15,24)(16,29)(19,23)(20,27)(21,33)(25,28)(31,35)(34,36)Polycyclic presentation:
2^9|a^2=h,c^2=efh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296332);