Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142540 |
| 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×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 624 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)(3,13,17,30)(4,19,11,5)(6,7,20,26)(9,22,29,15)(10,25,23,21)(14,24,32,28)(16,31,18,27);(1,3)(2,9)(4,13)(5,22)(6,18)(7,24)(8,17)(10,14)(11,30)(12,29)(15,19)(16,20)(21,27)(23,32)(25,31)(26,28);(1,4,20,25)(2,10,26,5)(3,14,16,22)(6,21,8,11)(7,19,12,23)(9,27,28,30)(13,29,31,24)(15,17,32,18)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72277942 |
| maximal normal subgroup of | #72277942 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49785);