Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| 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 | |
712 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,13,16,30)(4,14,18,27)(5,10,23,28)(6,12)(7,11)(8,25)(9,17,20,19)(15,31,26,21)(22,29,24,32);(1,3,5,20,8,26,24,9)(2,4,10,21,25,18,32,13)(6,16,22,14,7,15,23,27)(11,19,28,30,12,17,29,31);(1,4)(2,9)(3,10)(5,21)(6,17)(7,19)(8,18)(11,14)(12,27)(13,24)(15,28)(16,29)(20,25)(22,31)(23,30)(26,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fh,g^b=gi,h^b=hi,d^c=dghi,f^c=fghi,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237579, #72237583, #72237584, … |
| maximal normal subgroup of | #72237579, #72237583, #72237584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46646);