Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 602 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15)(7,16)(8,21,14,10)(9,13,25,23)(12,19)(17,31)(18,28)(20,27)(22,29,30,24)(26,33)(32,35)(34,36);(5,7,10,18,11,20,21,19)(6,12,14,27,15,28,8,16)(9,17,24,32,25,34,29,33)(13,26,30,36,23,35,22,31);(1,2,3,4)(5,8,10,6,11,14,21,15)(7,12,19,16,20,28,18,27)(9,23,24,22,25,13,29,30)(17,31,33,35,34,36,32,26);(5,9,21,29,11,25,10,24)(6,13,14,30,15,23,8,22)(7,17,19,33,20,34,18,32)(12,26,27,36,28,35,16,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290147);