Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149761 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11502 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C22×#120 | C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 855 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,8)(3,9,10,16,11,6,12,14)(5,7,13,15);(1,3)(2,6)(4,12)(5,11)(7,9)(8,14)(10,13)(15,16);(6,9)(14,16)Polycyclic presentation:
2^9|a^2=e,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=ef,g^b=gi,e^d=eh,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72216449, #72216452, #72216457, … |
| maximal normal subgroup of | #72216449, #72216452, #72216457, … |
| maximal quotient of | #72236998, #72237008, C2×#149761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56957);