Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #265163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3274 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(10,15)(11,17)(12,21)(13,22)(19,20)(23,24);(2,7)(3,4)(6,8)(9,10,13,15,14,12,22,21)(11,18,19,24,20,16,17,23);(1,2,5,8)(3,7,4,6)(9,11,14,20)(10,16,12,18)(13,19,22,17)(15,23,21,24);(2,8)(6,7)(9,12,22,15,14,10,13,21)(11,16,17,24,20,18,19,23)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 172377);