Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,20)(13,23)(16,28)(18,29)(21,27)(24,31)(25,32)(26,35)(30,36)(33,34);(5,7)(6,11,14,25)(8,16)(9,17)(10,18)(12,26,19,24)(13,15,22,29)(20,30)(21,31)(23,32)(27,36,33,28)(34,35);(5,8)(6,12,14,19)(7,16)(9,21)(10,20)(11,24,25,26)(13,27,22,33)(15,28,29,36)(17,31)(18,30)(23,34)(32,35);(2,4)(8,20)(12,19)(16,30)(21,34)(24,26)(27,33)(28,36)(31,35)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310558);