Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2366 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1004 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)(7,18,22,31)(8,24)(9,15)(10,16)(11,28)(12,17)(13,21,32,23)(14,25)(19,29,20,33)(26,34)(27,36,35,30);(5,7,10,23,12,22,11,21)(6,13,16,31,17,32,28,18)(8,19,9,27,26,20,25,35)(14,29,24,36,15,33,34,30);(5,8,11,25,12,26,10,9)(6,14,16,24,17,15,28,34)(7,19,21,35,22,20,23,27)(13,29,31,36,32,33,18,30);(5,9,10,26,12,25,11,8)(6,15,16,34,17,14,28,24)(7,20,21,27,22,19,23,35)(13,30,18,33,32,36,31,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=ei,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263541);