Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #60827 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #117 ◁ #3510 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,4,10)(3,13,15,29)(5,9,21,28)(6,12,18,30)(7,23,25,20)(8,26,27,14)(11,19,31,22)(16,24,17,32);(1,3)(2,7)(4,17)(5,20)(6,14)(8,9)(10,29)(11,13)(12,24)(15,21)(16,22)(18,23)(19,26)(25,28)(27,30)(31,32);(1,3)(2,8,12,13)(4,15)(5,16)(6,14)(7,11,24,9)(10,27,30,29)(17,21)(18,26)(19,23)(20,22)(25,31,32,28)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,e^c=ei,f^d=fhi,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59317);