Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2075 |
| 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 | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,12,16,14)(7,18,21,35)(8,9)(10,26)(13,30)(15,28,23,17)(19,34)(20,36,25,22)(24,27)(32,33);(1,2)(3,4)(5,6,10,28,11,16,26,17)(7,19,22,29,21,34,36,31)(8,23,9,12,24,15,27,14)(13,18,30,25,33,35,32,20);(2,4)(5,7,8,18,10,22,9,25,11,21,24,35,26,36,27,20)(6,13,12,29,17,32,23,19,16,33,14,31,28,30,15,34)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239851, #72239861, #72240857, … |
| maximal normal subgroup of | #72239851, #72239861, #72240857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56309);