Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149799 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | #11509 |
| prime radical | #3449 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C42 ◁ #2921 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,6,10,23,29,7,13,22,32,24,12,15,26)(3,14,16,30,18,8,25,9,19,27,4,20,17,28,21,31);(1,3)(2,8)(4,15)(5,21)(6,18)(7,19)(9,29)(10,27)(11,30)(12,14)(13,28)(16,23)(17,24)(20,32)(22,25)(26,31);(1,4)(2,9,13,31)(3,15)(5,18)(6,21)(7,16)(8,26,28,29)(10,20,12,30)(11,14,32,27)(17,22)(19,23)(24,25)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dh,e^b=efhi,f^b=fi,d^c=di,e^c=ei,e^d=ehi,f^e=fi38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56995);