Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #122108 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1446 |
| 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 | |
828 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 138 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)(11,19)(12,23,24,37)(13,17)(14,18)(15,21)(16,25,30,38)(20,33,35,40)(22,32)(26,31)(27,39,29,34)(28,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,29,30,27)(17,28,31,32)(23,33,37,40)(24,35)(25,34)(26,36)(38,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16,18,30)(11,20)(13,25)(14,24)(15,27,19,29)(17,23,31,37)(21,35)(22,34)(26,38)(28,40,32,33)(36,39)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fh,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29305);