Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,14)(9,25)(10,26)(11,16)(12,20)(13,23)(15,22)(18,30)(19,33)(21,32)(24,31)(27,28)(29,36)(34,35);(7,18,19,35)(8,9)(10,22)(12,28,29,32)(13,31)(14,25)(15,26)(17,34,33,30)(20,21,36,27)(23,24);(5,7,10,27,11,19,22,21)(6,12,15,34,16,29,26,30)(8,18,23,36,24,35,9,20)(13,32,14,33,25,28,31,17)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=dh,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250461, #72250651, #72250656, … |
| maximal normal subgroup of | #72250461, #72250651, #72250656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55991);