Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| 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 | |
994 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)(7,18,23,33)(8,24)(9,14)(10,15)(11,16)(12,17)(13,21,30,22)(19,35,20,28)(25,32)(26,31)(27,34,36,29);(5,7,11,21,12,23,10,22)(6,13,16,18,17,30,15,33)(8,19,25,36,26,20,9,27)(14,28,24,34,32,35,31,29);(5,8,10,9,12,26,11,25)(6,14,16,24,17,32,15,31)(7,19,22,27,23,20,21,36)(13,28,18,34,30,35,33,29);(5,9,10,26,12,25,11,8)(6,14,15,31,17,32,16,24)(7,20,21,27,23,19,22,36)(13,29,18,28,30,34,33,35)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=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263542);