Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397240 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,24,9,14,23,13)(7,15,29,30,18,11,17,26)(8,19,22,12,21,34,32,28)(16,31,36,25,20,35,33,27);(5,7,10,17,9,18,23,29)(6,11,13,26,14,15,24,30)(8,20,22,36,21,16,32,33)(12,27,19,31,28,25,34,35);(5,8,9,21)(6,12)(7,16)(10,22,23,32)(11,25,15,27)(13,19)(14,28)(17,33)(18,20)(24,34)(26,35,30,31)(29,36);(1,2,3,4)Polycyclic presentation:
2^9|a^2=efi,b^2=f,c^2=e,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 320281);