Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(7,16,18,35)(8,20)(9,24)(11,27,28,17)(12,30)(13,21)(14,25)(19,36,26,22)(23,31)(29,33,34,32);(1,2,4,3)(5,6)(7,17,18,27)(8,21)(9,25)(10,15)(11,16,28,35)(12,31)(13,20)(14,24)(19,32,26,33)(22,29,36,34)(23,30);(1,3,4,2)(5,7,9,26,10,18,24,19)(6,11,14,34,15,28,25,29)(8,22,20,16,23,36,31,35)(12,32,30,27,13,33,21,17)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72240831, #72250013, #72250032, … |
| maximal normal subgroup of | #72240831, #72250013, #72250032, … |
| maximal quotient of | #72240837 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55987);