Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8301 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 848 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(1,2,6,12,7,11,21,28,8,13,4,10,5,19,18,29)(3,14,15,31,16,9,23,26,17,27,22,30,20,25,32,24);(1,3)(2,9)(4,15)(5,20)(6,22)(7,16)(8,17)(10,24)(11,27)(12,26)(13,25)(14,19)(18,23)(21,32)(28,30)(29,31);(1,4,7,18,8,6,5,21)(2,10,11,29,13,12,19,28)(3,15,16,23,17,22,20,32)(9,24,27,31,25,26,14,30)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72292724, C2×#129358 |
| maximal normal subgroup of | #72292724, C2×#129358 |
| maximal quotient of | C2×#129358 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36554);