Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#388 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,14,11,16,9,15)(7,17,19,12,21,28,20,29)(8,22,26,13,24,31,25,30)(18,27,33,36,23,35,34,32);(1,3,4,2)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,27)(14,28)(15,17)(16,29)(18,24)(22,32)(25,33)(26,34)(30,35)(31,36);(2,3)(5,8,9,25,11,24,10,26)(6,13,14,31,16,30,15,22)(7,18,19,33,21,23,20,34)(12,27,17,32,29,35,28,36);(8,24)(13,30)(18,23)(22,31)(25,26)(27,35)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265536);