Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249719 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,6)(5,9)(7,11)(8,14)(12,15)(13,16)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,26,28);(2,6)(4,14)(7,10)(8,9)(11,12)(15,16)(17,18,22,20)(19,23,27,26)(21,29,31,24)(25,28,32,30);(1,2)(3,7)(4,8)(5,9)(6,14)(10,11)(12,16)(13,15)(17,19,22,27)(18,23,20,26)(21,30,31,28)(24,32,29,25);(1,3,5,13)(2,7,9,15)(4,12,14,11)(6,10,8,16)(17,20,22,18)(19,26,27,23)(21,24,31,29)(25,30,32,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gh,e^d=eh,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 157230);