Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12102 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,20,24,31,30,32,29,23,26,21,25,22,19,27,28);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(17,19)(18,22)(20,25)(21,24)(23,30)(26,31)(27,28)(29,32);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(17,18,23,26)(19,22,30,31)(20,28,21,29)(24,32,25,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=gi,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44662);