Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152215 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | |
431 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 431 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:
(1,2,5,11)(3,13,15,23)(4,18,19,27)(6,9,21,10)(7,12,20,26)(8,22,24,14)(16,25,31,28)(17,29,30,32);(1,3,4,14,7,17,6,16)(2,8,9,13,12,25,18,29)(5,15,19,22,20,30,21,31)(10,23,26,28,27,32,11,24);(1,2)(3,8)(4,9)(5,10)(6,18)(7,12)(11,21)(13,14)(15,23)(16,29)(17,25)(19,26)(20,27)(22,28)(24,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,f^c=fgi,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, 59411);