Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126214 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7778 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #31231 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,13)(4,16,18,32)(5,20)(6,10)(7,12)(8,15)(9,14,27,31)(11,21)(17,29,19,24)(22,28,30,26)(23,25);(1,3)(2,8,12,13)(4,14)(5,21)(6,23)(7,15)(9,24,27,29)(10,20,11,25)(16,26,32,28)(17,30)(18,31)(19,22);(1,4)(2,9)(3,14)(5,22)(6,17)(7,18)(8,24)(10,26)(11,28)(12,27)(13,29)(15,31)(16,25)(19,21)(20,32)(23,30)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,h^a=hi,d^b=dfi,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi,e^d=ei38 conjugacy classes — view the full character table.
| maximal subgroup of | #72219580, #72226635, #72226637 |
| maximal normal subgroup of | #72219580, #72226635, #72226637 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33411);