Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1936 |
| 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 | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(7,16)(8,21)(9,24)(10,23)(13,17)(14,31)(15,22)(18,20)(19,36)(25,29)(26,27)(28,33)(30,35);(5,6,9,12,23,26,21,31,11,15,8,22,10,14,24,27)(7,17,18,34,33,13,28,29,20,35,16,32,19,30,36,25);(5,7,8,18,23,33,24,28,11,20,9,16,10,19,21,36)(6,13,12,25,26,35,31,34,15,30,22,29,14,17,27,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321511, #72321585, #72321592, … |
| maximal normal subgroup of | #72321511, #72321585, #72321592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54135);