Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×C22⋊C16 |
| prime radical | C22×D4 | C16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,18,14,19,24,32,15,17,25,30,27,31,37,39,13,16,23,28,26,29,36,38,11,20,21,33,22,34,35,40);(2,6)(3,4)(7,8)(10,16)(12,23)(17,20)(19,29)(21,25)(24,36)(31,34)(35,37);(1,3,5,4)(2,7,8,6)(9,11,13,15)(10,17,16,20)(12,21,23,25)(14,22,26,27)(18,30,28,33)(19,31,29,34)(24,35,36,37)(32,39,38,40)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=fi,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,c^b=cfi,d^b=dfi,e^b=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57302);