Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500010 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62818 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,16)(11,17)(12,15,20,22)(14,19)(18,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,5)(3,7)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(1,3,4,7)(2,6,5,8);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=gi,d^2=h,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72310724, #72310729, #72310732, … |
| maximal normal subgroup of | #72310724, #72310729, #72310732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424034);