Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12217 |
| commutator, Frattini, cosocle | #379 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
348 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 348 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:
(1,2)(3,11)(4,15,18,8)(5,9)(6,10)(7,13)(12,29,16,27)(14,28)(17,24,19,25)(20,26)(21,30,23,32)(22,31);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,23,15,21)(9,11)(10,22)(17,30,19,32)(20,31)(24,27,25,29)(26,28);(1,4)(2,8)(3,12)(5,19)(6,18)(7,21)(9,25)(10,15)(11,27)(13,30)(14,16)(17,20)(22,23)(24,26)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,c^b=ch,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,f^c=fi,e^d=ehi,g^d=gi26 conjugacy classes — view the full character table.
| maximal subgroup of | #72227525, #72227526, #72227533, … |
| maximal normal subgroup of | #72227525, #72227526, #72237736, … |
| maximal quotient of | #72227523, #72227525, #72227526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58587);