Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478251 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17137 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
973 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 973 conjugacy classes of subgroups — too many to draw. Only its 168 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)(3,7,14,24)(4,8)(5,19)(6,10)(9,17)(11,21)(12,23,30,22)(13,26,31,32)(15,29,20,25)(16,28)(18,27);(1,3)(2,7,10,24)(4,13)(5,20)(6,14)(8,26,27,32)(9,22,28,23)(11,29,19,25)(12,16)(15,21)(17,30)(18,31);(1,4)(2,8)(3,12,14,30)(5,16)(6,18)(7,22,24,23)(9,11)(10,27)(13,20,31,15)(17,21)(19,28)(25,26,29,32);(1,4)(2,9,10,28)(3,13)(5,17)(6,18)(7,23,24,22)(8,19,27,11)(12,15)(14,31)(16,21)(20,30)(25,26,29,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237071, #72237088, … |
| maximal normal subgroup of | #72237066, #72237071, #72237088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385447);