Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1852 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
275 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 275 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,7,4,5,10)(3,11,13,8,15,16,14,12)(18,23)(19,22)(20,27)(24,32)(25,31)(26,29)(28,30);(1,3,6,13,7,15,5,14)(2,8,9,16,4,12,10,11)(17,18,21,25)(19,24,22,30)(20,28,29,32)(23,26,31,27);(1,4)(2,7)(3,12)(5,9)(6,10)(8,15)(11,13)(14,16)(17,19,21,22)(18,24,25,30)(20,26,29,27)(23,28,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh,e^c=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43649);