Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2571 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
612 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 186 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:
(2,4)(5,6,9,12,11,16,10,15)(7,17,20,13,21,27,19,29)(8,22,25,32,23,30,26,14)(18,34,36,33,35,28,24,31);(6,12,16,15)(8,23)(13,17,29,27)(14,30,32,22)(18,35)(24,36)(25,26)(28,31,34,33);(5,7,9,20,11,21,10,19)(6,13,15,17,16,29,12,27)(8,24,25,18,23,36,26,35)(14,31,30,28,32,33,22,34);(1,2,3,4)(5,8,10,26,11,23,9,25)(6,14,12,22,16,32,15,30)(7,18,20,36,21,35,19,24)(13,28,17,33,29,34,27,31)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383199);