Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #221184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #61087 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1048 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)(3,6)(4,7)(5,8)(9,10,14,18,15,11,13,21)(12,17,22,19,23,20,24,16);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,19,23,16)(13,18,14,21)(17,22,20,24);(2,7)(3,4)(6,8)(9,12)(10,16)(11,19)(13,22)(14,24)(15,23)(17,21)(18,20);(2,8)(6,7)(9,12)(10,17)(11,20)(13,24)(14,22)(15,23)(16,21)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128465);