Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149777 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#353 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 978 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,15,25,32)(7,16,18,31,20,27,34,35)(8,12,21,29,22,13,24,30)(11,23,26,17,28,33,36,19);(5,7)(6,11)(8,17)(9,23)(10,20)(12,16)(13,27)(14,31)(15,28)(18,21)(19,22)(24,34)(25,33)(26,30)(29,36)(32,35);(2,4)(5,8,10,22)(6,12,15,13)(7,17,20,19)(9,21,25,24)(11,16,28,27)(14,29,32,30)(18,33,34,23)(26,31,36,35)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,f^b=fgh,g^b=gi,f^d=fhi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59239);