Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499054 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,13,30,8,16,9,17,12,18,25,14,11,19,10,27)(7,20,28,33,29,34,22,32,24,31,26,15,21,36,23,35);(1,2)(3,4)(6,14,18,30)(8,11)(9,13)(10,25)(15,31,33,20)(16,17,19,27)(21,29)(22,28)(23,26)(32,36,35,34);(2,4)(5,7)(6,15)(8,21)(9,26)(10,28)(11,29)(12,24)(13,23)(14,20)(16,32)(17,34)(18,33)(19,35)(22,25)(27,36)(30,31);(5,8,12,11)(6,16,18,19)(7,21,24,29)(9,25,10,13)(14,27,30,17)(15,32,33,35)(20,36,31,34)(22,28,23,26)Polycyclic presentation:
2^9|a^2=eh,b^2=efh,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224928, #72224942, #72226816, … |
| maximal normal subgroup of | #72224928, #72224942, #72226816, … |
| maximal quotient of | #72224920 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406743);