Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1229 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)(3,4)(5,6,10,16,12,17,11,25)(7,15,19,28,20,27,9,13)(8,14,22,31,24,33,23,32)(18,30,34,36,35,29,21,26);(1,3,4,2)(5,7)(6,13)(8,21)(9,10)(11,19)(12,20)(14,29)(15,25)(16,27)(17,28)(18,23)(22,35)(24,34)(26,32)(30,33)(31,36);(2,3)(5,8)(6,14)(7,18)(9,21)(10,22)(11,23)(12,24)(13,26)(15,30)(16,31)(17,33)(19,34)(20,35)(25,32)(27,29)(28,36);(2,3)(5,9,11,20,12,19,10,7)(6,15,25,13,17,27,16,28)(8,21,23,35,24,34,22,18)(14,30,32,26,33,29,31,36)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337605);