Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | D4⋊C16 |
| prime radical | C23 | #350 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2)(3,4)(5,6,9,17,12,16,37,18,13,19,32,47,8,29,31,51,14,20,38,50,33,48,40,52,11,41,36,53,34,64,10,39)(7,21,23,43,26,56,61,68,27,55,58,67,30,15,24,42,28,45,35,65,59,44,62,54,25,63,60,66,22,46,57,49);(2,4)(5,7,10,24,34,22,36,58,11,25,40,61,33,59,38,23,14,28,31,57,8,30,32,60,13,27,37,62,12,26,9,35)(6,15,18,43,29,63,50,42,41,56,39,66,16,45,47,68,20,46,52,65,64,55,17,49,19,44,51,67,48,21,53,54)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286543, #72286999, #72287225, … |
| maximal normal subgroup of | #72286543, #72286999, #72287225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1239);