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 | |
1070 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)(6,12)(7,16)(8,19)(9,10)(13,27)(14,31)(15,26)(17,33)(18,23)(20,24)(21,22)(25,28)(29,36)(30,34);(1,2,3,4);(5,6)(7,13)(8,20)(9,14)(10,24)(11,15)(12,19)(16,28)(17,34)(18,30)(21,26)(22,31)(23,29)(25,32)(27,33)(35,36);(5,7,11,18)(6,13,15,30)(8,17,22,33)(9,23,10,16)(12,25,26,36)(14,29,24,28)(19,32,21,35)(20,34,31,27)Polycyclic presentation:
2^9|b^2=e,d^2=i,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, 419096);