Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,12,9,15)(7,17,20,28,21,13,18,26)(8,14,24,31,22,25,23,30)(19,29,34,36,33,27,32,35);(1,2)(3,4)(6,12)(7,18)(9,10)(13,26)(14,25)(17,28)(19,32)(20,21)(23,24)(27,35)(29,36)(33,34);(2,4)(5,7,11,21)(6,13,12,17)(8,19,22,33)(9,20,10,18)(14,29,25,27)(15,26,16,28)(23,34,24,32)(30,36,31,35);(5,8)(6,14)(7,19)(9,23)(10,24)(11,22)(12,25)(13,27)(15,30)(16,31)(17,29)(18,32)(20,34)(21,33)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287029);