Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,4,6,5,8,3,7)(9,10)(11,21)(12,17,26,33)(13,28)(14,20)(15,31)(16,19)(18,22)(23,37,35,36)(24,30)(25,39,29,40)(27,38,32,34);(2,6)(3,4)(7,8)(9,11,22,20,15,24,30,31,14,18,21,10,16,28,13,19)(12,23,37,33,27,39,25,38,26,35,36,17,32,40,29,34);(2,6)(3,4)(7,8)(9,12)(10,17)(11,23)(13,29)(14,26)(15,27)(16,32)(18,35)(19,34)(20,33)(21,36)(22,37)(24,39)(25,30)(28,40)(31,38)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=eh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45784);