Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10046 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
517 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 517 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,11)(4,16)(5,22)(6,9)(7,13)(8,20)(10,19)(12,27)(14,30)(15,29)(17,28)(18,24)(21,26)(23,32)(25,31);(4,17)(5,19)(7,23)(8,26)(10,22)(11,27)(12,15)(13,31)(16,28)(18,32)(20,21)(29,30);(1,3)(2,7)(4,18)(5,12)(6,14)(8,27)(9,23)(10,24)(11,16)(13,20)(15,19)(17,32)(21,31)(22,25)(26,30)(28,29)Polycyclic presentation:
2^9|d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214311, #72214352, #72214466, … |
| maximal normal subgroup of | #72214311, #72214352, #72214466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50398);