Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1437 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,10,14,11,15,9,24)(7,12,19,27,18,28,17,34)(8,20,22,13,23,30,25,31)(16,32,33,26,21,36,35,29);(1,2)(3,4)(5,7)(6,12)(8,21,23,16)(9,17)(10,19)(11,18)(13,29,31,26)(14,27)(15,28)(20,36,30,32)(22,35,25,33)(24,34);(1,3,4,2)(5,8)(6,13)(7,16)(9,25)(10,22)(11,23)(12,26)(14,30)(15,31)(17,35)(18,21)(19,33)(20,24)(27,36)(28,29)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30938);