Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,10,15)(7,16,20,34)(8,21,24,13)(9,12,25,14)(11,26,27,18)(17,32,19,33)(22,31,23,30)(28,36,29,35);(5,7,8,18,10,20,24,26)(6,11,13,16,15,27,21,34)(9,19,22,36,25,17,23,35)(12,29,30,33,14,28,31,32);(6,12)(7,17)(9,25)(11,27)(13,30)(14,15)(18,36)(19,20)(21,31)(22,23)(26,35)(32,33)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312426, #72312428, #72312435, … |
| maximal normal subgroup of | #72312426, #72312428, #72312435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55475);