Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397950 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1946 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)(8,12)(9,22)(10,14)(11,17)(13,23)(16,30)(18,29)(19,28)(20,35)(21,27)(24,34)(25,33)(26,31)(32,36);(5,7,9,17,10,18,23,33)(6,11,13,15,14,25,22,29)(8,19,20,34,21,16,31,32)(12,24,26,28,27,36,35,30);(5,8,10,21)(6,12,14,27)(7,16)(9,20,23,31)(11,24)(13,26,22,35)(15,28)(17,32)(18,19)(25,36)(29,30)(33,34);(2,4)Polycyclic presentation:
2^9|b^2=e,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316311, #72316334, #72316343, … |
| maximal normal subgroup of | #72316311, #72316334, #72316343, … |
| maximal quotient of | #72316881, #72316999, #72317027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305147);