Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,19,15,20,26,35,16,11,17,22,31,23,33,37,14,18,13,27,25,34,28,39,30,21,29,24,32,36,40,38);(2,5,7,6)(9,11,13,22,15,23,28,37,16,18,29,27,31,34,40,39,14,21,12,24,25,36,26,38,30,10,17,19,32,20,33,35)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef,c^b=ci,d^b=di152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1231);