Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152179 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12759 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #61110 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 826 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,12,7,13,22,28)(3,14,17,23,19,29,31,26)(4,11,21,8,20,24,6,10)(9,18,25,32,27,15,30,16);(2,8)(4,5)(6,21)(9,23)(10,13)(11,24)(14,29)(15,18)(16,17)(20,22)(26,27)(31,32);(1,3)(2,9)(4,16)(5,17)(6,15)(7,19)(8,23)(10,26)(11,14)(12,25)(13,27)(18,21)(20,32)(22,31)(24,29)(28,30)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=egh,f^b=fghi,g^b=gi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59376);