Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94188 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C4×C4⋊Q8 | C22 |
| prime radical | C23 | #342 |
| LCS | G ▷ C2×C4⋊C4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4×C4⋊Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18,21,25,22,19,20,28)(23,26,30,27,32,29,31,24);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19,22,18)(20,24,21,27)(23,26,32,29)(25,31,28,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1458);