Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #268283 |
| 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 | |
979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8,4,3,5,7)(10,15,17,18)(11,19,20,21);(5,6)(7,8)(9,10)(11,16)(12,18)(13,15)(14,17)(19,22)(20,24)(21,23);(1,3,4,2)(5,7,6,8)(10,15)(12,13)(16,22)(17,18)(19,21)(23,24);(5,6)(7,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=fh,c^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=gi,e^d=ei,f^d=fhi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 175484);