Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7523 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1404 |
| commutator | #125 | C22×C4 |
| Frattini, cosocle | #491 | C23 |
| socle | C2 | C2×#1404 |
| prime radical | #18990 | C2 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #3135 ◁ G | |
| LECS | G ▷ #491 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(1,2,5,10,6,12,3,9)(4,15,17,28,16,25,32,23)(7,11,14,27,21,30,20,29)(8,19,24,22,26,13,31,18);(2,8)(4,16)(7,21)(9,23)(10,28)(11,25)(12,26)(13,19)(14,20)(15,30)(24,27)(29,31);(1,3,6,5)(2,9,12,10)(4,13,16,19)(7,14,21,20)(8,23,26,28)(11,27,30,29)(15,31,25,24)(17,18,32,22)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=dh,e^b=efi,f^b=fhi,g^b=gh,e^d=egh,f^d=fh,g^d=gi,f^e=fhi,g^e=gi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60685);