Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,13)(11,15,18,22)(12,16,20,21)(17,24,19,23);(1,2,4,5)(3,6,7,8)(9,11,14,18)(10,15,13,22)(12,19,20,17)(16,24,21,23);(2,5)(3,7)(9,12)(10,16)(11,17)(13,21)(14,20)(15,23)(18,19)(22,24);(1,3)(2,6)(4,7)(5,8)(10,13)(12,20)(15,22)(17,19);(2,5)(3,7)(9,13,14,10)(11,15,18,22)(12,21,20,16)(17,23,19,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303431, #72303501, #72303545, … |
| maximal normal subgroup of | #72303431, #72303501, #72303545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502980);