Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12705 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10725 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1670 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1670 conjugacy classes of subgroups — too many to draw. Only its 200 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:
(1,2,5,10,7,12,6,11)(3,13,16,29,15,28,17,30)(4,18,20,27,19,9,21,25)(8,22,24,14,23,32,26,31);(1,3,6,17,7,15,5,16)(2,8,11,24,12,23,10,26)(4,14,20,32,19,31,21,22)(9,13,25,30,18,28,27,29);(4,19)(9,18)(13,28)(14,31)(20,21)(22,32)(25,27)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ei,f^b=fh,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317357, #72317369, #72317390, … |
| maximal normal subgroup of | #72317357, #72317369, #72317390, … |
| maximal quotient of | C2×#146478 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53674);