Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470225 |
| 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 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22);(1,3)(2,5)(4,7)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22);(10,18)(15,24)(16,17)(22,23)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238109, #72238110, #72256551, … |
| maximal normal subgroup of | #72238109, #72238110, #72256551, … |
| maximal quotient of | #72257634, #72316510, #72317700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396941);