Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133259 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8522 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C4×M16 | C23 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
611 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 611 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:
(1,2,4,10,5,11,18,29,6,8,17,25,20,9,26,27)(3,12,13,30,14,7,21,23,15,24,31,28,19,22,16,32);(1,3,6,15)(2,7,8,22)(4,16,17,21)(5,19,20,14)(9,24,11,12)(10,28,25,30)(13,18,31,26)(23,27,32,29);(2,8)(7,22)(9,11)(12,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,f^c=fh,f^d=fi,f^e=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72306758, #72306763, #72306765, … |
| maximal normal subgroup of | #72306758, #72306763, #72306765, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40457);