Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1112 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)(7,18)(8,22)(9,16)(10,15)(11,17)(12,19)(13,20)(14,25)(21,33)(23,34)(24,28)(26,32)(27,31)(29,36)(30,35);(1,2,3,4);(5,7)(6,12)(8,23)(9,20)(10,27)(11,21)(13,28)(14,31)(15,29)(16,34)(17,30)(18,22)(19,25)(24,35)(26,36)(32,33);(5,7)(6,13)(8,19)(9,27)(10,20)(11,21)(12,28)(14,30)(15,33)(16,18)(17,31)(22,34)(23,25)(24,36)(26,35)(29,32)Polycyclic presentation:
2^9|b^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419095);