Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,17,11,16)(7,9,24,28,14,26,19,25)(8,20,23,30,22,31,27,13)(18,32,34,21,29,35,33,36);(5,7)(6,9)(8,18,22,29)(10,24)(11,19)(12,14)(13,21,30,36)(15,28)(16,25)(17,26)(20,35,31,32)(23,34,27,33);(5,8,10,27,12,22,11,23)(6,13,15,31,17,30,16,20)(7,18,19,34,14,29,24,33)(9,21,25,35,26,36,28,32);(2,4)(5,9,12,26)(6,14,17,7)(8,21,22,36)(10,25,11,28)(13,29,30,18)(15,24,16,19)(20,34,31,33)(23,32,27,35)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342025);