Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1343 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | #2280 | C22 |
| socle | C2 | #7250 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #2280 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,5,9,6,10,18,24)(3,11,13,25,14,26,27,32)(4,8,16,22,17,23,29,31)(7,19,20,15,21,28,30,12);(1,3)(2,7,10,21)(4,15)(5,13)(6,14)(8,11)(9,20,24,30)(12,17)(16,28)(18,27)(19,29)(22,25)(23,26)(31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=i,b^a=bc,c^a=cf,e^a=efgi,h^a=hi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338380, #72338381, #72338382, … |
| maximal normal subgroup of | #72338380, #72338381, #72338382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1489);