Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130050 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 559 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 40 points:
(2,5,6,7)(9,10,14,19,15,20,29,36)(11,21,24,37,23,34,28,39)(12,18,25,35,13,17,27,33)(16,30,31,26,32,38,40,22);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,22)(13,26)(14,28)(15,23)(17,21)(18,34)(19,31)(20,32)(24,29)(25,38)(27,30)(33,37)(35,39)(36,40);(9,12,15,13)(10,17)(11,22,23,26)(14,25,29,27)(16,21)(18,20)(19,33)(24,30,28,38)(31,37)(32,34)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37371);