Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2088 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 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)(5,6,10,18,8,25,9,16,12,17,26,14,11,15,27,32)(7,19,23,33,21,13,22,30,20,34,36,29,24,31,35,28);(5,7,8,21,12,20,11,24)(6,13,15,19,17,31,25,34)(9,22,26,36,27,35,10,23)(14,28,16,29,18,30,32,33);(2,4)(6,14,17,18)(7,20)(8,11)(9,10)(13,28,31,30)(15,32,25,16)(19,33,34,29)(22,36)(23,35)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250750, #72250757, #72250785, … |
| maximal normal subgroup of | #72250750, #72250757, #72250785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56067);