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:
(2,7,9,17,11,21,10,27)(3,12)(4,16)(5,13)(8,22,24,23,26,32,25,20)(14,31)(15,29)(18,28);(1,2)(3,8)(4,17)(5,9)(6,11)(7,19)(10,13)(12,24)(14,32)(15,26)(16,27)(18,22)(20,31)(21,30)(23,28)(25,29);(1,3,13,29,6,15,5,12)(2,8,10,25,11,26,9,24)(4,18,30,28,16,14,19,31)(7,20,17,22,21,23,27,32)Polycyclic presentation:
2^9|a^2=e,c^2=ghi,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, 57916);