Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13,15,10)(8,16,18,11)(9,12)(14,20)(17,19);(1,2)(3,4)(5,7)(6,10)(8,17)(9,15)(11,19)(12,13)(14,18)(16,20);(2,4)(5,8)(6,11)(7,14,15,17)(9,18)(10,19,13,20)(12,16);(10,13)(19,20)Polycyclic presentation:
2^9|a^2=h,c^2=ef,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225669, #72225679, #72225680, … |
| maximal normal subgroup of | #72225669, #72225679, #72225680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291452);