Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 1288 conjugacy classes of subgroups — too many to draw. Only its 314 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,13,10,14,8,12)(7,15,18,16,19,20,17,11);(5,7,8,18,10,19,9,17)(6,11,13,20,14,16,12,15);(1,3,4,2)(5,6,10,14)(7,16)(8,12,9,13)(11,19)(15,17)(18,20);(7,17)(8,9)(11,20)(12,13)(15,16)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=egi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,h^a=hi,c^b=cef,d^b=de,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284632, #72284684, #72284738, … |
| maximal normal subgroup of | #72284632, #72284684, #72284738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264569);