Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152219 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12793 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #3018 | C22 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2,6,11,7,4,18,20)(3,13,16,29,17,27,32,25)(5,12,21,10,22,9,23,19)(8,24,26,28,14,30,31,15);(1,3,5,15,7,17,22,28)(2,8,9,13,4,14,12,27)(6,16,21,24,18,32,23,30)(10,25,11,26,19,29,20,31);(1,4)(2,7)(3,14)(5,12)(6,10)(8,17)(9,22)(11,21)(13,28)(15,27)(16,25)(18,19)(20,23)(24,26)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=ei,f^c=fg,g^c=gi,h^c=hi,f^e=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, 59417);