Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1461 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 1461 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,7)(3,10)(8,11)(9,15)(12,13)(14,16);(1,2)(3,11)(4,8)(5,14)(6,9)(7,10)(12,15)(13,16);(3,12)(7,11)(10,13)(15,16)Polycyclic presentation:
2^9|d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72209860, #72209889, #72213349, … |
| maximal normal subgroup of | #72209860, #72209889, #72213349, … |
| maximal quotient of | #72213317, #72213323, #72213434, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49732);