Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12170 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1598 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #9041 |
| prime radical | #33530 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,7,14,26,16,6,12,9,17,25,32,8,15,5,13)(3,11,19,10,23,28,18,29,21,27,24,30,20,31,4,22);(1,3,8,23,9,21,26,20)(2,10,16,22,17,30,15,29)(4,5,19,25,18,6,24,7)(11,12,31,32,27,13,28,14);(1,4)(2,11)(3,5)(6,21)(7,20)(8,19)(9,18)(10,12)(13,30)(14,29)(15,28)(16,31)(17,27)(22,32)(23,25)(24,26)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72286542 |
| maximal normal subgroup of | #72286542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44598);