Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135286 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1823 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,15,26,30,11,16,8,22,25,35,23,12)(7,17,18,34,20,33,24,13,21,29,19,31,28,36,27,32);(1,2)(3,4)(6,12,16,30)(8,23)(9,26)(10,25)(13,29,32,17)(14,15,22,35)(18,24)(19,27)(20,28)(31,33,34,36);(2,4)(5,7)(6,13,16,32)(8,24)(9,27)(10,28)(11,21)(12,29,30,17)(14,33,22,36)(15,34,35,31)(18,23)(19,26)(20,25)Polycyclic presentation:
2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,e^b=eh,g^b=gh,h^b=hi,d^c=dh,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224839, #72224942, #72226813, … |
| maximal normal subgroup of | #72224839, #72224942, #72226813, … |
| maximal quotient of | #72224836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56570);