Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #273179 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3,6,8)(9,10)(11,19)(12,16)(13,20)(14,17)(15,18)(21,23)(22,24);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,21)(14,16)(15,22)(17,24)(18,23)(19,20);(2,6)(4,7)(9,12)(10,16)(11,20)(13,19)(14,24)(15,23)(17,22)(18,21);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,24)(14,19)(15,23)(16,20)(17,21)(18,22)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,g^c=gh,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 180378);