Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135507 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
725 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 725 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)(3,11,14,7)(4,16,18,8)(5,9)(6,10)(12,19)(13,27,29,20)(15,24,25,21)(17,32,26,22)(23,28,31,30);(1,3,12,25)(2,7,19,24)(4,13,26,29)(5,14,6,15)(8,20,32,27)(9,11,10,21)(16,31,22,23)(17,28,18,30);(2,8,9,22)(4,17)(7,20,11,23)(10,16,19,32)(13,28)(18,26)(21,31,24,27)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=eg,f^c=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72235243, #72235247, #72235255, … |
| maximal normal subgroup of | #72235243, #72235247, #72235255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42730);