Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| 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 | C23×Q8 | 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 ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1 | |
1336 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1336 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:
(1,2,5,11,7,12,20,28)(3,13,16,29,17,22,31,32)(4,9,18,25,6,10,19,26)(8,21,23,15,24,30,27,14);(1,3,6,15,7,17,4,14)(2,8,9,22,12,24,10,13)(5,16,18,21,20,31,19,30)(11,27,26,32,28,23,25,29);(1,4,7,6)(2,9,12,10)(3,14,17,15)(5,18,20,19)(8,22,24,13)(11,25,28,26)(16,21,31,30)(23,32,27,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72317470, #72317494, #72317501, … |
| maximal normal subgroup of | #72317470, #72317494, #72317501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35218);