Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150924 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11903 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | #532 | D4 |
| 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 | |
241 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 241 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:
(2,7,9,20,10,21,25,15)(3,11)(4,14)(5,17)(8,22,23,19,24,30,32,31)(12,28)(13,27)(16,26);(1,2)(3,8)(4,15,14,20)(5,9)(6,10)(7,18,21,29)(11,23)(12,22,16,30)(13,24)(17,25)(19,26,31,28)(27,32);(1,3,5,11,6,13,17,27)(2,8,9,23,10,24,25,32)(4,16,18,28,14,12,29,26)(7,19,15,22,21,31,20,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=ghi,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=efgh,h^b=hi,d^c=dhi,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, 58120);