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, socle | C23 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(5,6,8,7)(10,15)(12,13)(16,24)(17,22)(18,19)(20,23);(5,6,8,7)(9,10)(11,18)(12,22)(13,15)(14,17)(16,20)(19,23)(21,24);(1,2,4,3)(5,7,8,6)(9,10)(11,19)(12,15)(13,22)(14,17)(16,21)(18,23)(20,24);(1,3,4,2)(9,11)(10,16)(12,23)(13,20)(14,21)(15,24)(17,19)(18,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,e^b=ei,d^c=di,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264322);