Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
478 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)(8,10)(9,22)(13,29)(14,33)(15,26)(17,31)(18,24)(19,20)(21,25)(27,34)(28,30)(32,35);(5,6,8,12,11,15,10,26)(7,17,18,35,20,30,27,29)(9,23,21,33,25,36,22,14)(13,24,28,16,32,34,31,19);(5,7,8,18,11,20,10,27)(6,13,12,28,15,32,26,31)(9,24,21,16,25,34,22,19)(14,29,23,17,33,35,36,30)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322185, #72322199, #72322215, … |
| maximal normal subgroup of | #72322185, #72322199, #72322215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55967);