Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,25,23,8,10,14,11,15,27,22,9,16,24,31)(7,17,26,30,32,29,21,33,19,18,34,36,20,13,28,35);(1,2)(3,4)(5,7,9,20,11,19,23,32)(6,13,8,17,15,29,16,18)(10,21,12,26,24,28,27,34)(14,30,22,33,31,36,25,35);(2,4)(5,8,9,15,11,16,23,6)(7,18,20,13,19,17,32,29)(10,14,12,22,24,31,27,25)(21,36,26,35,28,30,34,33);(7,19)(13,29)(17,18)(20,32)(21,28)(26,34)(30,36)(33,35)Polycyclic presentation:
2^9|a^2=fh,b^2=e,c^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250521, #72250526, #72250536, … |
| maximal normal subgroup of | #72250521, #72250526, #72250536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406636);