Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,11,16,12,9,10,15)(7,13,21,29,20,18,19,28)(8,22,25,23,24,31,26,14)(17,32,34,30,33,36,35,27);(5,7)(6,13)(8,17,24,33)(9,18)(10,19)(11,21)(12,20)(14,30,23,27)(15,28)(16,29)(22,36,31,32)(25,34,26,35);(5,8,10,25,12,24,11,26)(6,14,16,31,9,23,15,22)(7,17,21,35,20,33,19,34)(13,27,28,32,18,30,29,36);(5,9,12,6)(7,18,20,13)(8,23,24,14)(10,15,11,16)(17,30,33,27)(19,28,21,29)(22,26,31,25)(32,35,36,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efg,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293613);