Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138620 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(2,5)(3,6)(7,8)(9,10,13,18,28,37,29,38,15,20,12,17,14,19,30,39)(11,21,23,33,31,36,26,16,25,32,22,40,24,35,27,34);(1,2,6,8,4,7,3,5)(9,11)(10,16)(12,26)(13,27)(14,31)(15,25)(17,35)(18,36)(19,40)(20,34)(21,38)(22,29)(23,30)(24,28)(32,39)(33,37);(9,11)(10,16)(12,27)(13,26)(14,31)(15,25)(17,36)(18,35)(19,40)(20,34)(21,39)(22,30)(23,29)(24,28)(32,38)(33,37)Polycyclic presentation:
2^9|a^2=f,b^2=dfghi,d^2=ghi,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=eh,f^c=fhi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309414, #72309426, #72309432, … |
| maximal normal subgroup of | #72309414, #72309426, #72309432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45837);