Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
642 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,17,22,18,21,27,28)(15,16,24,25,23,19,20,26);(5,6,7,9,8,10,11,12)(13,15,17,20)(14,19,28,25)(16,22,26,21)(18,23,27,24);(2,4)(13,16,18,19)(14,20,21,24)(15,22,23,28)(17,26,27,25)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45153);