Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #121711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1445 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
676 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,7,4,6)(5,8)(9,10)(11,15,20,30)(12,22)(13,18)(14,17)(16,24)(19,29,37,40)(21,31,26,35)(23,32,38,39)(25,36)(27,33)(28,34);(1,3)(2,6)(4,5)(7,8)(9,11)(10,15,17,30)(12,23)(13,26)(14,20)(16,32,22,39)(18,35,34,31)(19,25)(21,28)(24,38)(27,37)(29,36,40,33);(1,4,5,3)(2,7,8,6)(9,12,14,24)(10,16)(11,19,20,37)(13,27,28,25)(15,29)(17,22)(18,36)(21,38,26,23)(30,40)(31,39)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=deh,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,d^c=di,e^c=ei,f^c=fi,g^c=ghi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28908);