Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485039 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
3348 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 3348 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 4 generators on 16 points:
(1,2)(3,6)(4,7)(5,9)(8,10)(11,13)(12,14)(15,16);(2,6)(3,10)(7,8)(9,13)(11,16)(14,15);(3,11)(10,16);(2,7)(6,8)(9,14)(13,15)Polycyclic presentation:
2^9|e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,e^c=eh,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213827, #72216457, #72231046, … |
| maximal normal subgroup of | #72213827, #72216457, #72231046, … |
| maximal quotient of | #72213804, #72213808, #72237721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392235);