Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18)(19,26,28,22)(20,25,29,23)(21,24)(27,30)(31,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,22)(20,27)(21,28)(23,30)(24,26)(25,31)(29,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(20,29)(22,30)(24,25)(26,31)(27,32)Polycyclic presentation:
2^9|a^2=f,c^2=gi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ei,f^c=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72261158, #72261175, #72261206 |
| maximal normal subgroup of | #72261158, #72261175, #72261206 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41126);