Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(6,12,15,26)(7,18,20,24)(8,23)(9,25)(10,28)(13,31)(14,17,30,16)(19,22,27,21)(29,36)(33,34);(5,6,10,16,11,15,28,17)(7,13,21,33,20,34,22,31)(8,14,23,12,25,30,9,26)(18,35,27,36,24,32,19,29);(1,3,4,2)(5,7,10,21,11,20,28,22)(6,13,16,33,15,34,17,31)(8,24,23,19,25,18,9,27)(12,29,30,35,26,36,14,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250812, #72250819, #72250827, … |
| maximal normal subgroup of | #72250812, #72250819, #72250827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54482);