Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111405 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1078 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2)(3,7,8,6)(4,5)(9,10,14,20,15,19,13,18)(11,21,24,16,25,33,23,32)(12,26,28,36,30,17,29,35)(22,34,38,31,39,37,27,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,25)(10,16,19,32)(12,27,30,38)(13,24,14,23)(17,34,26,37)(18,33,20,21)(22,28,39,29)(31,35,40,36);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,26)(20,36)(21,37)(23,38)(24,27)(25,39)(32,40)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18604);