Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150854 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11880 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | #24287 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(2,7,9,16,11,20,10,26)(3,12)(4,15)(5,17)(8,21,23,22,25,30,24,19)(13,29)(14,28)(27,31);(1,2)(3,8)(4,16)(5,9)(6,11)(7,18)(10,17)(12,23)(13,30)(14,25)(15,26)(19,29)(20,32)(21,31)(22,27)(24,28);(1,3,5,12,6,14,17,28)(2,8,9,23,11,25,10,24)(4,13,18,27,15,31,32,29)(7,19,26,30,20,22,16,21)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,c^b=cf,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dh,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58050);