Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12948 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,15,11,16,10,26)(7,17,20,33,19,13,21,29)(8,22,24,30,25,12,23,14)(18,27,36,28,35,31,34,32);(6,12,16,22)(8,23)(9,10)(13,27,17,31)(14,15,30,26)(18,34)(20,21)(24,25)(28,29,32,33)(35,36);(1,3,4,2)(5,7)(6,13,16,17)(8,18)(9,20)(10,21)(11,19)(12,27,22,31)(14,28,30,32)(15,29,26,33)(23,34)(24,36)(25,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250198, #72250210, #72250219, … |
| maximal normal subgroup of | #72250198, #72250210, #72250219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54429);