Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2058 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
690 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)(5,6,9,16,11,15,10,27)(7,17,20,34,19,13,21,31)(8,22,25,33,26,12,23,14)(18,29,36,30,24,28,35,32);(6,12,15,22)(8,23)(9,10)(13,29,17,28)(14,16,33,27)(18,35)(20,21)(24,36)(25,26)(30,31,32,34);(5,7)(6,13,15,17)(8,24)(9,20)(10,21)(11,19)(12,28,22,29)(14,32,33,30)(16,31,27,34)(18,26)(23,36)(25,35)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231484, #72231541, #72231546, … |
| maximal normal subgroup of | #72231484, #72231541, #72231546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56667);