Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7318 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1304 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | C2×#1304 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1356 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1356 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,20,27)(3,13,15,28,16,22,29,31)(4,9,17,25,5,10,19,26)(8,21,23,18,24,30,32,14);(1,3)(2,8,12,24)(4,14)(5,18)(6,15,20,29)(7,16)(9,22,10,13)(11,23)(17,21,19,30)(25,31)(26,28)(27,32);(1,4,7,5)(2,9,12,10)(3,14,16,18)(6,17,20,19)(8,22,24,13)(11,25,27,26)(15,21,29,30)(23,31,32,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=i,b^a=bd,d^a=dg,f^a=fhi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72258290, #72258301, #72258308, … |
| maximal normal subgroup of | #72258290, #72258301, #72258308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35226);