Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149894 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2122 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | #13083 |
| prime radical | #23994 | C2 |
| LCS | G ▷ C2×#41 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,12,7,9,21,27)(3,13,15,30,17,28,32,23)(4,18,19,26,6,11,22,10)(8,16,24,31,25,14,29,20);(1,3)(2,8,18,30,9,25,11,23)(4,16)(5,20)(6,14)(7,17)(10,24,27,13,26,29,12,28)(15,19)(21,31)(22,32);(1,4,7,6)(2,9)(3,14,17,16)(5,19,21,22)(10,26)(11,18)(12,27)(15,20,32,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59889);