Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2075 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,17,11,14,9,16)(7,18,19,29,22,12,21,30)(8,23,25,32,26,13,24,15)(20,31,33,27,34,36,35,28);(1,2)(3,4)(5,7)(6,12)(8,20)(9,19)(10,21)(11,22)(13,31)(14,18)(15,28)(16,30)(17,29)(23,36)(24,33)(25,35)(26,34)(27,32);(2,4)(6,13,14,23)(7,19)(8,24)(9,10)(12,27,18,28)(15,17,32,16)(20,34)(21,22)(25,26)(29,36,30,31)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239850, #72239866, #72240857, … |
| maximal normal subgroup of | #72239850, #72239866, #72240857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55808);