Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498821 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,8)(9,17)(10,19)(11,31)(12,15)(13,18)(14,24)(16,28)(20,34)(21,30)(22,26)(23,33)(25,29)(27,36)(32,35);(1,2)(3,4)(5,7)(6,8)(9,20)(10,29)(11,14)(12,33)(13,22)(15,23)(16,32)(17,34)(18,26)(19,25)(21,27)(24,31)(28,35)(30,36);(1,3,4,2)(5,8)(6,14)(7,19)(9,23)(10,30)(11,32)(12,34)(13,26)(15,35)(16,20)(17,21)(18,29)(22,31)(24,28)(25,27)(33,36);(5,9,13,12)(6,15,18,17)(7,20,22,33)(8,23,26,34)(10,27,11,28)(14,35,29,21)(16,31,36,19)(24,30,25,32)Polycyclic presentation:
2^9|c^2=e,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250186, #72250204, #72250215, … |
| maximal normal subgroup of | #72250186, #72250204, #72250215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406270);