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)(7,12)(8,22)(9,15)(10,14)(11,16)(13,23)(17,27)(18,35)(19,29)(20,28)(21,30)(24,31)(25,32)(26,33)(34,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,27)(14,28)(15,29)(16,30)(17,23)(22,35)(24,33)(25,34)(26,31)(32,36);(1,3,4,2)(5,8)(6,13)(7,17)(9,25)(10,23)(11,24)(12,26)(14,31)(15,22)(16,32)(18,19)(20,33)(21,34)(27,29)(28,36)(30,35);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,27)(14,22)(15,31)(16,32)(17,19)(20,34)(21,33)(26,29)(28,35)(30,36)Polycyclic presentation:
2^9|c^2=e,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, 301860);