Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188761 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,11,17,24,32,9,15,22,31,23,14,29,30)(7,13,18,28,19,27,33,36,8,12,20,26,21,25,34,35);(5,7)(6,12)(8,9)(10,20)(11,19)(13,15)(14,27)(16,28)(17,25)(18,22)(21,23)(24,34)(26,31)(29,33)(30,35)(32,36);(1,2,3,4)(5,8,9,7)(6,13,15,12)(10,20,22,18)(11,21,23,19)(14,25,17,27)(16,28,31,26)(24,34,29,33)(30,35,32,36);(6,14,15,17)(7,8)(10,22)(12,25,13,27)(16,32,31,30)(19,21)(24,29)(26,36,28,35)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=ci,d^b=di,g^b=gi,d^c=di,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401387);