Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,20)(8,28)(9,15)(10,17)(11,16)(12,30)(13,19)(14,23)(18,31)(21,24)(22,34)(25,35)(26,29)(27,36)(32,33);(5,7)(6,14)(8,29)(9,22)(10,24)(11,23)(12,32)(13,27)(15,33)(16,34)(17,20)(18,21)(19,35)(25,28)(26,31)(30,36);(2,4)(5,8,10,18,31,17,28,6,13,16,12,15,9,30,11,19)(7,21,23,29,22,35,32,36,27,33,25,34,26,14,24,20);(5,9,13,31)(6,15,19,18)(7,22,27,26)(8,17,16,30)(10,11,12,28)(14,33,35,21)(20,34,36,29)(23,32,25,24)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320223, #72320262, #72320443, … |
| maximal normal subgroup of | #72320223, #72320262, #72320443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406381);