Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478195 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
835 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 835 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,17,11,15)(7,18,21,13,23,32,22,33)(8,9,26,14,27,30,25,29)(19,28,24,34,36,20,31,35);(5,7,10,21,12,23,11,22)(6,13,15,18,17,33,16,32)(8,24,25,19,27,31,26,36)(9,28,14,34,30,20,29,35);(5,8,10,26,12,27,11,25)(6,14,15,9,17,29,16,30)(7,19,22,31,23,36,21,24)(13,28,32,34,33,20,18,35);(2,4)(5,9,10,29,12,30,11,14)(6,8,15,26,17,27,16,25)(7,20,22,35,23,28,21,34)(13,31,32,36,33,24,18,19)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386588);