Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499952 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3205 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23930 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
967 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 967 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 4 generators on 32 points:
(1,2)(3,12)(4,17)(5,10)(6,22)(7,11)(8,15)(9,21)(13,26)(14,29)(16,28)(18,32)(19,27)(20,25)(23,30)(24,31);(3,13,14,16)(4,18)(8,23,24,27)(9,25)(12,28,29,26)(15,19,31,30)(17,32)(20,21);(1,3)(2,8)(4,19)(5,16)(6,13)(7,14)(9,26)(10,27)(11,24)(12,17)(15,21)(18,30)(20,31)(22,23)(25,28)(29,32);(4,18)(9,25)(12,29)(15,31)(17,32)(19,30)(20,21)(26,28)Polycyclic presentation:
2^9|b^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219593, #72226637, #72226639, … |
| maximal normal subgroup of | #72219593, #72226637, #72226639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407148);