Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15930 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
937 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 937 conjugacy classes of subgroups — too many to draw. Only its 164 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 — 4 generators on 32 points:
(1,2)(3,7,13,21)(4,8)(5,16)(6,10)(9,18)(11,22,26,19)(12,20,17,29)(14,23)(15,24)(25,31,27,32)(28,30);(1,3)(2,7,10,21)(4,11)(5,17,18,12)(6,13)(8,22,24,19)(9,20)(14,27,28,25)(15,26)(16,29)(23,32)(30,31);(1,4)(2,8)(3,11)(5,14)(6,15)(7,19)(9,23)(10,24)(12,25)(13,26)(16,30)(17,27)(18,28)(20,31)(21,22)(29,32);(1,5)(2,9,10,16)(3,12)(4,14)(6,18)(7,20,21,29)(8,23,24,30)(11,25)(13,17)(15,28)(19,31,22,32)(26,27)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,d^2=g,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237737, #72237740, #72237746, … |
| maximal normal subgroup of | #72237737, #72237740, #72237746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356286);