Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,15,17)(11,19,20,22)(12,21,13,16)(14,23,24,18);(2,5)(3,7)(9,11)(10,12)(13,17)(14,21)(15,20)(16,24)(18,22)(19,23);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,11,17,20)(14,19,24,22)(16,23,21,18);(2,5)(6,8)(9,13)(10,16)(11,18)(12,15)(14,19)(17,21)(20,23)(22,24);(1,3,4,7)(2,6,5,8)(10,17)(11,20)(16,21)(18,23)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050010);