Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,9)(8,21)(10,29)(11,16)(12,17)(13,19)(14,25)(15,27)(18,30)(20,28)(22,31)(23,32)(24,34)(26,33)(35,36);(1,3,4,2)(5,7,10,19,12,20,11,15)(6,13,16,9,17,27,29,28)(8,22,24,35,26,32,25,18)(14,31,33,30,34,23,21,36);(5,8,11,25,12,26,10,24)(6,14,16,33,17,34,29,21)(7,18,19,22,20,35,15,32)(9,23,13,30,28,31,27,36);(5,9,10,27,12,28,11,13)(6,15,16,7,17,19,29,20)(8,23,24,36,26,31,25,30)(14,32,33,18,34,22,21,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275609);