Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×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 | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
696 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)(5,6,11,16)(7,19,17,9)(8,22)(10,15,24,18)(12,14,30,21)(13,28)(20,31)(23,25)(26,33)(27,35)(29,34)(32,36);(2,4)(5,7,12,15,10,21,19,6,11,17,30,18,24,14,9,16)(8,23,27,33,28,35,25,34,29,32,31,13,26,20,36,22);(5,8,24,26,11,29,10,28)(6,13,18,22,16,33,15,34)(7,20,21,23,17,35,14,32)(9,25,30,27,19,36,12,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239801, #72239814, #72239826, … |
| maximal normal subgroup of | #72239801, #72239814, #72239826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54633);