Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #444620 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15784 |
| 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 | |
1121 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1121 conjugacy classes of subgroups — too many to draw. Only its 184 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:
(1,2,5,11,7,8,6,20)(3,12,13,23,15,9,21,25)(4,16,18,22,17,30,19,10)(14,28,27,26,29,32,31,24);(2,8)(3,13)(5,6)(9,23)(12,25)(14,27)(15,21)(16,30)(18,19)(24,28)(26,32)(29,31);(1,3,5,13,7,15,6,21)(2,9,20,23,8,12,11,25)(4,14,18,27,17,29,19,31)(10,24,30,32,22,26,16,28);(1,4)(2,10,8,22)(3,14)(5,18)(6,19)(7,17)(9,24,12,26)(11,16,20,30)(13,27)(15,29)(21,31)(23,32,25,28)Polycyclic presentation:
2^9|a^2=g,c^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72216217, #72216948, … |
| maximal normal subgroup of | #72215912, #72216217, #72216948, … |
| maximal quotient of | C2×#444620, #72252169, #72314126 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351816);