Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,18)(12,17)(13,19)(14,20)(15,21)(16,23)(22,24);(9,11,14,23,15,12,13,22)(10,16,19,18,21,24,20,17);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,15)(13,23)(14,22)(16,19)(18,21)(20,24);(1,4,5,3)(2,6,7,8)(9,11)(10,18)(12,15)(13,22)(14,23)(16,20)(17,21)(19,24);(9,13,15,14)(10,19,21,20)(11,22,12,23)(16,18,24,17)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213480, #72213751, #72219560, … |
| maximal normal subgroup of | #72213480, #72213751, #72219560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359416);