Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141878 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1988 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9869 |
| prime radical | C23 | #421 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,13,17,30,15,31,18,27)(4,10,14,29)(5,9,7,11)(8,22,23,19,25,16,24,32)(20,26,21,28);(1,3,14,32,7,17,20,22,6,15,4,19,5,18,21,16)(2,8,10,27,11,23,28,13,12,25,29,30,9,24,26,31);(2,9,12,11)(3,15)(4,14)(8,23,25,24)(10,28,29,26)(13,27,31,30)(17,18)(20,21)Polycyclic presentation:
2^9|a^2=f,b^2=defg,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=ch,d^b=dg,e^b=ei,f^b=fgi,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49074);