Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | C4×#99 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
709 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)(5,6,11,8,10,14,25,21)(7,15,19,30,18,29,33,36)(9,22,24,34,23,13,28,20)(12,26,16,31,27,35,32,17);(5,7)(6,12)(8,16)(9,17,23,31)(10,18)(11,19)(13,15,22,29)(14,27)(20,30,34,36)(21,32)(24,26,28,35)(25,33);(2,4)(5,8,11,14,10,21,25,6)(7,16,19,27,18,32,33,12)(9,20,24,22,23,34,28,13)(15,17,30,26,29,31,36,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,g^a=ghi,d^b=dh,h^b=hi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54518);