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 | #60371 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2139 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)(7,19)(8,25)(9,28)(10,16)(11,18)(12,17)(13,22)(14,27)(15,29)(20,31)(21,35)(23,32)(24,33)(26,30)(34,36);(1,2)(3,4)(5,7)(6,13)(8,26)(9,21)(10,22)(11,23)(12,24)(14,34)(15,31)(16,19)(17,32)(18,33)(20,27)(25,35)(28,30)(29,36);(1,3,4,2)(5,8,10,9)(6,14,16,15)(7,20,22,36)(11,29,12,27)(13,30,19,35)(17,28,18,25)(21,23,26,24)(31,33,34,32);(5,9,10,8)(6,15,16,14)(7,21,22,26)(11,29,12,27)(13,31,19,34)(17,28,18,25)(20,23,36,24)(30,33,35,32)Polycyclic presentation:
2^9|c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301861);