Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94486 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7324 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | #2308 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11)(4,8,15,21)(5,17)(6,10)(7,12,19,25)(9,18)(13,23)(14,24)(16,30,29,22)(20,26,31,27)(28,32);(1,3,5,13)(2,7,9,20,10,19,17,31)(4,12,16,26)(6,14,18,28)(8,11,22,23,21,24,30,32)(15,25,29,27)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,e^2=fi,b^a=bc,c^a=cf,e^a=egi,f^a=fi,c^b=cg,d^b=dfi,g^b=gh,h^b=hi,d^c=di,e^c=eh,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72237837, #72237840, #72237844, … |
| maximal normal subgroup of | #72237837, #72237840, #72237844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1683);