Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2196 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #12958 |
| prime radical | #3018 | C22 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,6,11,7,13,24,12)(3,9,17,29,18,30,32,26)(4,19,15,8,16,27,20,28)(5,21,22,25,14,10,23,31);(1,3,6,17,7,18,24,32)(2,8,11,27,13,28,12,19)(4,14,15,23,16,5,20,22)(9,25,29,10,30,31,26,21);(1,4)(2,9)(3,14)(5,18)(6,15)(7,16)(8,25)(10,27)(11,29)(12,26)(13,30)(17,23)(19,21)(20,24)(22,32)(28,31)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gh,d^b=di,e^b=eghi,h^b=hi,d^c=dg,e^c=ei,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59663);