Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11)(7,15)(8,20)(9,23)(14,26)(17,33)(18,19)(21,22)(24,28)(25,30)(27,36)(32,35);(1,2)(3,4)(5,6,9,11,10,14,23,26)(7,16,18,28,19,12,15,24)(8,13,21,25,22,29,20,30)(17,32,34,36,33,27,31,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,19)(11,24)(13,27)(14,16)(15,23)(20,31)(21,34)(22,33)(25,35)(26,28)(29,32)(30,36);(5,8)(6,13)(7,17)(9,21)(10,22)(11,25)(12,27)(14,29)(15,31)(16,32)(18,34)(19,33)(20,23)(24,35)(26,30)(28,36)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301488);