Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1418 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)(7,15,18,27)(8,12)(9,13)(10,14)(11,17,23,21)(16,29,30,36)(19,25)(20,26)(22,28)(24,31,34,33)(32,35);(2,4)(5,7)(6,11)(8,16)(9,21)(10,18)(12,24)(13,27)(14,23)(15,28)(17,22)(19,30)(20,33)(25,34)(26,36)(29,35)(31,32);(2,4)(8,19)(11,23)(12,25)(15,27)(16,30)(20,32)(26,35)(31,33);(5,8,9,20,10,19,22,32)(6,12,13,26,14,25,28,35)(7,16,21,33,18,30,17,31)(11,24,27,36,23,34,15,29)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265888);