Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151297 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12158 |
| commutator, Frattini, cosocle | C2×M32 | C23 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
255 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,8)(4,9)(5,20)(6,11)(7,13)(10,14)(12,15)(16,22)(17,25)(18,23)(19,29)(21,28)(24,30)(26,32)(27,31);(2,8)(3,14)(4,6)(5,15)(9,22)(10,12)(11,23)(13,25)(16,32)(17,31)(18,30)(19,21)(20,24)(26,28)(27,29);(2,9,13,11)(8,22,25,23)(10,24,27,26)(12,20,29,28)Polycyclic presentation:
2^9|c^2=g,d^2=fh,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,g^a=gi,h^a=hi,d^b=df,e^b=egi,f^b=fh,h^b=hi,d^c=dg,e^d=ei38 conjugacy classes — view the full character table.
| maximal subgroup of | #72320907, #72320958, #72320994, … |
| maximal normal subgroup of | #72320907, #72320958, #72320994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58493);