Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142969 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #3490 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,8,6)(3,14,25,31)(4,11,19,24)(5,21,20,29)(7,12,9,13)(10,26,23,16)(15,18,30,17)(22,27,32,28);(1,3)(2,10)(4,16)(5,22)(6,23)(7,17)(8,25)(9,18)(11,15)(12,27)(13,28)(14,21)(19,26)(20,32)(24,30)(29,31);(1,4,9,20)(2,11,13,29)(3,15,18,31)(5,8,19,7)(6,24,12,21)(10,26,28,22)(14,25,30,17)(16,27,32,23)Polycyclic presentation:
2^9|a^2=gi,c^2=i,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,h^b=hi,d^c=dh,f^c=fgi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72303990, #72304006, #72304010 |
| maximal normal subgroup of | #72303990, #72304006, #72304010 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50181);