Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 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)(7,17)(8,22,25,28)(9,13,27,24)(10,15)(11,16)(12,20)(14,29)(18,26,23,30)(19,34,35,33)(21,31)(32,36);(1,2)(3,4)(5,7,10,20,11,21,29,36)(6,12,14,17,16,32,15,31)(8,23,24,34,25,18,13,33)(9,26,22,35,27,30,28,19);(2,4)(5,8,10,24,11,25,29,13)(6,9,15,28,16,27,14,22)(7,18,20,33,21,23,36,34)(12,30,31,35,32,26,17,19);(5,9)(6,13)(7,19)(8,15)(10,28)(11,27)(12,18)(14,25)(16,24)(17,34)(20,30)(21,35)(22,29)(23,32)(26,36)(31,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296313);