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 | |
498 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)(5,6,10,16,12,18,31,15)(7,19,21,32,24,34,23,28)(8,25,27,20,29,22,26,13)(9,17,11,14,30,35,33,36);(2,4)(5,7,11,22,31,21,9,13,12,24,33,25,10,23,30,20)(6,8,17,28,16,26,14,34,18,29,35,32,15,27,36,19);(5,8,10,27,12,29,31,26)(6,13,16,25,18,20,15,22)(7,14,21,35,24,36,23,17)(9,19,11,32,30,34,33,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321342, #72321379, #72321421, … |
| maximal normal subgroup of | #72321342, #72321379, #72321421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54405);