Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12127 |
| commutator, Frattini, cosocle | #423 | C23 |
| prime radical | #10212 | C2 |
| LCS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 3 generators on 32 points:
(2,8)(4,15)(5,6)(9,25)(10,16)(11,24)(12,29)(13,20)(17,31)(18,21)(19,23)(22,26)(27,30)(28,32);(1,2,6,10,7,11,5,19)(3,9,20,27,14,28,13,26)(4,16,21,8,18,23,15,24)(12,30,17,25,31,22,29,32);(1,3)(2,9)(4,17)(5,13)(6,20)(7,14)(8,22)(10,27)(11,28)(12,15)(16,25)(18,29)(19,26)(21,31)(23,32)(24,30)Polycyclic presentation:
2^9|b^2=fh,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fi,g^a=gh,d^b=dfi,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=ehi,g^c=gi,e^d=eh,g^d=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72339771, #72339806, #72339807 |
| maximal normal subgroup of | #72339771, #72339806, #72339807 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58414);