Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470067 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,16)(11,20)(13,14)(18,19)(21,24)(22,23);(1,3)(2,5)(4,7)(6,8)(9,10,15,19)(11,12,21,17)(13,18,14,16)(20,22,24,23);(3,7)(5,8)(9,11)(10,17)(12,19)(13,20)(14,24)(15,21)(16,22)(18,23);(3,7)(5,8)(9,12,15,17)(10,11,19,21)(13,23,14,22)(16,20,18,24);(2,6)(5,8)(9,13,15,14)(10,18,19,16)(11,20,21,24)(12,22,17,23)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,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=dh,f^b=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311985, #72312199, #72312623 |
| maximal normal subgroup of | #72311985, #72312199, #72312623 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377272);