Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,13)(12,17)(14,18)(15,20)(16,19)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(1,3,4,2)(6,10)(7,14)(8,17)(12,13)(15,16)(18,20);(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29);(5,7,9,16)(6,11,13,19)(8,14,17,15)(10,18,12,20)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,f^c=fi,e^d=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134217);