Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,14,11,15,9,25)(7,16,20,27,19,32,18,12)(8,13,23,29,24,30,22,36)(17,31,34,28,21,35,33,26);(5,7)(6,12,15,27)(8,21,24,17)(9,18)(10,20)(11,19)(13,28)(14,16,25,32)(22,34,23,33)(26,30)(29,35)(31,36);(1,2,3,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,26)(14,29)(15,30)(16,31)(18,33)(19,21)(20,34)(25,36)(27,28)(32,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,f^b=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31127);