Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152666 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #23873 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3500 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 593 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:
(1,2,7,14)(3,16,19,28)(4,20,17,31)(5,22,9,15)(6,13,25,32)(8,27,24,12)(10,21,29,26)(11,23,30,18);(1,3,9,4)(2,10,13,31,15,30,27,28)(5,23)(6,21,8,18)(7,26)(11,22,16,12,29,14,20,32)(17,25)(19,24);(1,4,9,3)(2,11,15,29)(5,17,7,19)(6,18,8,21)(10,22,30,14)(12,28,32,31)(13,16,27,20)(23,25,26,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,e^b=ei,f^b=fg,g^b=gh,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59863);