Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94168 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7213 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | #2308 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(1,2,5,9)(3,11,13,26)(4,15,16,30)(6,10,19,25)(7,20,21,12)(8,23,24,17)(14,27,18,31)(22,29,32,28);(1,3)(2,7,10,22)(4,12)(5,18,19,13)(6,14)(8,11,15,27)(9,21)(16,20,23,29)(17,28)(24,31)(25,32)(26,30)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=ghi,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72237729, #72237731, #72237734, … |
| maximal normal subgroup of | #72237729, #72237731, #72237734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1364);