Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,19)(8,22,26,33)(9,16)(10,15)(11,14)(12,18)(13,25,32,28)(17,29)(20,30,34,27)(21,31)(23,24,36,35);(2,4)(5,7,11,15,10,14,19,6,12,16,31,17,29,21,9,18)(8,23,27,32,28,34,24,33,26,36,30,13,25,20,35,22);(5,8,10,28,12,26,29,25)(6,13,15,33,18,32,17,22)(7,20,21,36,16,34,14,23)(9,24,11,30,19,35,31,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,c^b=ch,e^b=ei,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321294, #72321321, #72321342, … |
| maximal normal subgroup of | #72321294, #72321321, #72321342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54628);