Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,10,17,26,15)(7,12,20,22,19,8,13,23)(9,14,25,32,24,30,36,31)(18,27,29,34,33,21,28,35);(1,2)(3,4)(5,7,10,19)(6,12,17,8)(9,18,24,33)(11,13,26,20)(14,27,30,21)(15,22,16,23)(25,28,36,29)(31,34,32,35);(2,4)(5,8)(6,13)(7,16)(9,21,24,27)(10,12)(11,23)(14,29,30,28)(15,19)(17,20)(18,32,33,31)(22,26)(25,35,36,34);(5,9)(6,14)(7,18)(8,21)(10,24)(11,25)(12,27)(13,28)(15,31)(16,32)(17,30)(19,33)(20,29)(22,34)(23,35)(26,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287468);