Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1442 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,16,19,28)(8,20)(9,14)(10,24)(11,15)(12,18,29,25)(13,22)(17,33,21,27)(23,30)(26,31)(32,34,36,35);(5,7)(6,12,15,29)(8,21,23,17)(9,18)(10,25)(11,19)(13,27)(14,16,24,28)(20,36)(22,35,26,34)(30,32)(31,33);(2,4)(5,8,9,22,11,23,10,26)(6,13,14,30,15,31,24,20)(7,17,18,34,19,21,25,35)(12,27,16,32,29,33,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,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, 30948);