Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144613 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2029 |
| 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 | |
565 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,8,24)(3,14,19,27)(4,9,7,13)(5,11,6,22)(10,26,25,15)(12,21,31,20)(16,32,18,30)(17,29,23,28);(2,9)(3,15,18,17)(5,20)(6,21)(10,27,29,32)(13,24)(14,28,30,25)(16,23,19,26);(1,3)(2,10)(4,19)(5,15)(6,23)(7,18)(8,16)(9,25)(11,30)(12,32)(13,29)(14,31)(17,21)(20,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=dg,b^2=di,e^2=g,f^2=gi,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,e^b=eh,f^b=fi,d^c=dg,e^c=eg,f^c=fgi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72266273, #72266277, #72324272, … |
| maximal normal subgroup of | #72266273, #72266277, #72324272, … |
| maximal quotient of | #72321522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51818);