Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1804 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8565 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
373 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(1,2,6,12,7,9,21,24,8,13,19,28,4,11,5,20)(3,14,16,10,15,25,31,23,18,30,32,26,17,29,22,27);(2,9,13,11)(5,21)(6,19)(10,23,26,27)(12,20,28,24)(14,29,30,25)(16,32)(22,31);(1,3)(2,10)(4,17)(5,22)(6,16)(7,15)(8,18)(9,23)(11,27)(12,25)(13,26)(14,20)(19,32)(21,31)(24,30)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41589);