Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134690 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(22,26,27,32)(23,29)(24,31)(28,35,34,33)(30,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,22,23,26,24,28,29,33,25,27,30,32,31,34,36,35);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318790, #72318792, #72318811, … |
| maximal normal subgroup of | #72318790, #72318792, #72318811, … |
| maximal quotient of | #72322537 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42938);