Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#354 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4⋊Q8 | C24 |
| socle | C2 | C22×#354 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
829 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 829 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 4 generators on 32 points:
(1,2,5,10,6,11,21,29)(3,12,15,30,13,24,31,28)(4,8,17,26,18,9,20,27)(7,22,25,16,23,19,32,14);(1,3,6,13)(2,7)(4,14,18,16)(5,19,21,22)(8,24)(9,12)(10,28)(11,23)(15,20,31,17)(25,27)(26,32)(29,30);(3,13)(7,23)(12,24)(14,16)(15,31)(19,22)(25,32)(28,30);(1,4,6,18)(2,8,11,9)(3,14,13,16)(5,17,21,20)(7,24,23,12)(10,26,29,27)(15,22,31,19)(25,28,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^b=eh,f^b=fgh,g^b=gi,f^e=fhi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419826);