Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12483 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1947 |
| prime radical | C23 | #477 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,11,15,9)(3,7,4,13,12,8,14,16)(18,24)(19,20)(21,22)(25,27)(26,32)(28,29)(30,31);(1,3,5,4,6,12,15,14)(2,7,10,13,11,8,9,16)(17,18)(19,24)(20,30)(21,26)(22,31)(23,27)(25,29)(28,32);(1,4,5,12,6,14,15,3)(2,8,9,13,11,7,10,16)(17,19,22,28,23,29,21,20)(18,25,31,30,27,24,26,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ei,d^2=gi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=dg,g^b=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47698);