Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533135 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4052 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)(3,8,6,4)(5,7);(1,3,5,6)(2,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22);(1,4)(2,6)(3,7)(5,8)(10,14)(15,21)(16,22)(23,24);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=h,e^2=h,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72339534, #72339537, #72339539, … |
| maximal normal subgroup of | #72339534, #72339537, #72339539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451243);