Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
611 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 611 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,7,14,9,12,4,20)(3,11,17,29,19,10,25,27)(5,24,23,32,8,15,22,31)(6,26,16,28,21,13,18,30);(1,3)(2,10)(4,21)(5,16)(6,7)(8,18)(9,19)(11,12)(13,14)(15,28)(17,22)(20,26)(23,25)(24,30)(27,31)(29,32);(1,4,5,22)(2,11,12,10)(3,6,16,25)(7,8,23,9)(13,20,26,14)(15,30,24,28)(17,19,21,18)(27,32,29,31)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fg,f^c=fgh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72238366, #72238368, #72238370, … |
| maximal normal subgroup of | #72238366, #72238368, #72238370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42671);