Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150719 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11829 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | #575 | C2×C4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2960 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,7,9,16,11,21,10,27)(3,12)(4,15)(5,18)(8,22,24,23,26,31,25,20)(13,30)(14,29)(17,28);(1,2)(3,8)(4,16)(5,9)(6,11)(7,19)(10,18)(12,24)(13,31)(14,26)(15,27)(17,22)(20,30)(21,32)(23,28)(25,29);(1,3,5,12,6,14,18,29)(2,8,9,24,11,26,10,25)(4,17,19,30,15,13,32,28)(7,20,27,31,21,23,16,22)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=eh,f^d=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57915);