Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2071 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 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)(7,20)(8,14)(9,27)(10,30)(11,18)(12,31)(13,19)(15,21)(16,29)(17,28)(22,35)(23,34)(24,32)(25,36)(26,33);(5,7,10,8,11,21,29,24,13,23,9,26,12,22,28,25)(6,14,17,20,31,36,27,35,19,33,16,34,18,32,30,15);(5,8,9,21,11,24,28,23,13,26,10,22,12,25,29,7)(6,15,16,14,31,20,30,36,19,35,17,33,18,34,27,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250079, #72250082, #72250085, … |
| maximal normal subgroup of | #72250079, #72250082, #72250085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56182);