Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| 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) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16,13,18,10,30)(7,19,21,32,23,36,20,25)(8,22,24,14,27,33,26,34)(9,17,12,31,29,35,28,15);(2,4)(5,7,12,22,11,20,29,34,13,23,28,33,10,21,9,14)(6,8,17,25,30,24,15,19,18,27,35,32,16,26,31,36);(5,8,11,24,13,27,10,26)(6,14,16,33,18,34,30,22)(7,15,21,17,23,31,20,35)(9,19,12,32,29,36,28,25)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=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250502, #72250511, #72250519, … |
| maximal normal subgroup of | #72250502, #72250511, #72250519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54401);