Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32976 |
| prime radical | #3481 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 783 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(1,2,7,10,6,11,4,16)(3,12,19,22,15,26,13,8)(5,9,20,23,18,24,17,29)(14,25,30,31,28,32,27,21);(1,3,4,13,6,15,7,19)(2,8,10,12,11,22,16,26)(5,14,17,27,18,28,20,30)(9,21,23,25,24,31,29,32);(2,9)(8,21)(10,23)(11,24)(12,25)(16,29)(22,31)(26,32);(5,18)(14,28)(17,20)(27,30)Polycyclic presentation:
2^9|a^2=eh,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,g^c=ghi,f^d=fh,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231599, #72231703, #72231705, … |
| maximal normal subgroup of | #72231599, #72231703, #72231705, … |
| maximal quotient of | #72257195, C2×#512038, #72313916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419234);