Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150449 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11761 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | #2010 | C4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(2,6,7,11,9,14,8,15)(3,10)(4,12);(1,2)(3,11)(4,7)(5,9)(6,13)(8,12)(10,15)(14,16);(3,10)(13,16)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dhi,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57645);