Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94483 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7324 |
| commutator | C25 | C42 |
| Frattini, cosocle | #2309 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #2309 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 305 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:
(1,2,4,9,6,8,14,21)(3,7,12,18,11,20,25,22)(5,16,15,28,17,10,24,23)(13,19,27,30,26,29,32,31);(1,3,5,13)(2,7,10,19,8,20,16,29)(4,12,15,27)(6,11,17,26)(9,22,23,31,21,18,28,30)(14,25,24,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=i,e^2=f,b^a=bc,c^a=cf,e^a=eg,f^a=fi,c^b=cg,d^b=df,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 | #72284788, #72284835, #72284849, … |
| maximal normal subgroup of | #72284788, #72284835, #72284849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1680);