Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131988 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#538 | C4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
809 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 809 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10,13,18,14,17,26,33)(11,19,22,34,21,15,27,29)(12,16,25,31,24,32,37,40)(20,28,35,38,23,30,36,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,23)(13,22)(14,21)(16,30)(17,19)(18,34)(20,24)(25,36)(26,27)(28,32)(29,33)(31,38)(35,37)(39,40);(3,8)(5,7)(9,12)(10,16)(11,20)(13,25)(14,24)(15,28)(17,32)(18,31)(19,30)(21,23)(22,35)(26,37)(27,36)(29,38)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39190);