Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,18,8,17,9,16,12,19,26,14,11,15,28,35)(7,20,24,30,22,13,27,32,25,31,36,29,21,33,23,34);(1,2)(3,4)(5,7,8,22,12,25,11,21)(6,13,15,20,19,33,17,31)(9,27,26,36,28,23,10,24)(14,34,16,29,18,32,35,30);(2,4)(6,14,19,18)(7,21)(8,11)(9,10)(13,29,33,30)(15,35,17,16)(20,34,31,32)(22,25)(23,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,d^2=i,e^2=h,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,e^b=ei,d^c=di,e^c=eh,f^c=fi,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249953, #72249959, #72249962, … |
| maximal normal subgroup of | #72249953, #72249959, #72249962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56065);