Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2398 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
996 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)(7,17,20,26)(8,14)(9,15)(10,13)(11,16)(12,19,27,24)(18,31,33,35)(21,29)(22,28)(23,30)(25,32,36,34);(5,7)(6,12,16,27)(8,18)(9,19)(10,24)(11,20)(13,26,15,17)(14,25,30,36)(21,32)(22,34)(23,33)(28,35,29,31);(2,4)(6,13,16,15)(12,17,27,26)(14,28,30,29)(25,31,36,35);(5,8,9,21,11,23,10,22)(6,14,15,29,16,30,13,28)(7,18,19,32,20,33,24,34)(12,25,17,31,27,36,26,35)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266032);