Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347226 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #63000 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(4,5)(7,8)(9,10,14,11)(12,19,20,24)(13,18,23,16)(15,22,17,21);(1,2,6,8)(3,4,7,5)(9,11,14,10)(12,17,20,15)(13,16,23,18)(19,22,24,21);(9,12)(10,15)(11,17)(13,22)(14,20)(16,24)(18,19)(21,23);(1,3)(2,4)(5,8)(6,7)Polycyclic presentation:
2^9|a^2=i,b^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,e^d=eh,g^d=gh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72271558 |
| maximal normal subgroup of | #72271558 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254488);