Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133080 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
997 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 997 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,4,10)(3,12,15,27)(5,8,18,24)(6,11,20,28)(7,21,22,13)(9,19,26,16)(14,25,17,30)(23,31,32,29);(2,7,11,23)(3,13)(4,17)(8,12,25,9)(10,27,28,26)(15,19)(16,29)(18,20)(21,31)(22,30,32,24);(2,8)(4,18)(7,12)(9,23)(11,25)(15,21)(17,20)(19,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fghi,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72238487, #72238488, #72238492, … |
| maximal normal subgroup of | #72238487, #72238488, #72238492, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40313);