Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 689 conjugacy classes of subgroups — too many to draw. Only its 58 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:
(1,2,8,12)(3,13,17,31)(4,7,11,6)(5,18,22,28)(9,25,26,15)(10,23,20,21)(14,30,29,19)(16,27,24,32);(1,3)(2,9)(4,14)(5,19)(6,16)(7,24)(8,17)(10,27)(11,29)(12,26)(13,18)(15,21)(20,32)(22,30)(23,25)(28,31);(1,4,7,12)(2,8,11,6)(3,14,24,26)(5,20,23,18)(9,17,29,16)(10,21,28,22)(13,30,32,15)(19,27,25,31)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72233390, #72233393, #72233397 |
| maximal normal subgroup of | #72233390, #72233393, #72233397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49751);