Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1244 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,21,23,27)(16,24,25,19)(17,20)(18,22)(26,28);(13,15)(14,18,20,26)(16,22)(17,23)(19,21,24,27)(25,28);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fgi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261213, #72261374, #72261381, … |
| maximal normal subgroup of | #72261213, #72261374, #72261381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36444);