Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
516 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)(6,13,16,28)(7,18,20,35)(8,9)(10,11)(14,29)(15,17,34,25)(19,21,24,26)(22,23)(27,36)(31,32);(5,6,10,17,12,16,11,25)(7,14,21,29,20,32,26,31)(8,15,22,28,23,34,9,13)(18,33,19,36,35,30,24,27);(1,3,4,2)(5,7,11,26,12,20,10,21)(6,14,25,31,16,32,17,29)(8,18,9,24,23,35,22,19)(13,27,34,30,28,36,15,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250554, #72250565, #72250570, … |
| maximal normal subgroup of | #72250554, #72250565, #72250570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54490);