Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8920 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,11,7,4,18,10)(3,12,15,26,17,29,16,31)(5,20,22,9,23,27,32,19)(8,24,25,30,13,21,28,14);(1,3,6,15,7,17,18,16)(2,8,11,25,4,13,10,28)(5,21,22,14,23,24,32,30)(9,26,27,29,19,31,20,12);(1,4)(2,7)(3,13)(5,20)(6,10)(8,17)(9,22)(11,18)(12,21)(14,26)(15,28)(16,25)(19,32)(23,27)(24,29)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=egi,g^b=gi,h^b=hi,d^c=dgi,e^c=eh,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43688);