Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16)(7,12,20,27)(8,21,24,28)(9,13,10,15)(14,22,29,23)(17,26,19,30)(18,32,34,35)(25,33,36,31);(1,2)(3,4)(5,7)(6,12)(8,18)(9,17)(10,19)(11,20)(13,26)(14,25)(15,30)(16,27)(21,32)(22,33)(23,31)(24,34)(28,35)(29,36);(1,3,4,2)(6,13)(7,17)(9,10)(14,28)(15,16)(18,31)(19,20)(21,29)(22,23)(26,30)(32,35)(33,34);(5,8)(6,14)(7,18)(9,23)(10,22)(11,24)(12,25)(13,28)(15,21)(16,29)(17,31)(19,33)(20,34)(26,35)(27,36)(30,32)Polycyclic presentation:
2^9|a^2=i,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301871);