Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×#97 | C22 |
| socle | C22 | #1280 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 338 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) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,18,14,20,28,36,15,17,26,33,13,19,27,35)(11,21,22,34,24,31,38,39,25,16,29,30,23,32,37,40);(2,5,7,6)(9,11,13,23,15,25,14,24)(10,16,19,31,17,21,20,32)(12,22,27,37,26,29,28,38)(18,34,35,40,33,30,36,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,g^a=gi,d^b=dg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240001, #72240004, #72240005, … |
| maximal normal subgroup of | #72240001, #72240004, #72240005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 499);