Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152742 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13088 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #25745 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3499 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,14,17,30)(4,11,19,24)(6,10,22,20)(7,13,8,23)(9,25,27,18)(15,26,21,32)(16,28,31,29);(1,2,8,24,7,13,5,20)(3,9,17,30,15,28,21,32)(4,10,22,12,6,11,19,23)(14,25,29,16,26,31,27,18);(1,3)(2,9)(4,18)(5,21)(6,16)(7,15)(8,17)(10,14)(11,26)(12,29)(13,28)(19,31)(20,32)(22,25)(23,27)(24,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59940);