Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128069 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1455 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1455 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(1,2,5,9)(3,11,13,26)(4,8,17,23)(6,10,18,24)(7,19,20,12)(14,25,27,32)(15,21,28,30)(16,22,29,31);(1,3)(2,7,10,21)(4,15)(5,13,18,27)(6,14)(8,11,22,25)(9,20)(12,16)(17,28,29,19)(23,26)(24,30)(31,32);(4,16)(8,22)(11,25)(12,15)(17,29)(19,28)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,h^a=hi,d^b=dh,e^b=egi,g^b=gi,d^c=di,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72226369, #72227300, C2×#128069, … |
| maximal normal subgroup of | #72226369, #72227300, C2×#128069, … |
| maximal quotient of | #72225899, #72225905, #72225907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35265);