Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1819 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 628 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)(3,14,16,24)(4,17,19,8)(5,13)(6,22,20,11)(7,12)(9,18,25,31)(10,21)(15,26,30,27)(23,28,32,29);(1,3)(2,8)(4,5)(6,15)(7,16)(9,11)(10,14)(12,17)(13,24)(18,20)(19,21)(22,27)(23,31)(25,28)(26,29)(30,32);(1,4,5,3)(2,9,10,25)(6,18,20,15)(7,19,21,16)(8,22,14,29)(11,17,28,24)(12,26,13,27)(23,30,32,31)Polycyclic presentation:
2^9|a^2=f,c^2=d,e^2=g,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, #72238369, … |
| maximal normal subgroup of | #72238366, #72238368, #72238369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42667);