Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #332983 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,14)(5,15)(6,16)(7,11)(9,12)(10,13)(17,18,21,25,23,26,22,32)(19,27,28,20,24,30,29,31);(1,3,7,6)(2,8,11,16)(4,12,5,13)(9,15,10,14)(17,18)(19,27)(20,29)(21,32)(22,25)(23,26)(24,30)(28,31);(1,4,7,5)(2,9,11,10)(3,12,6,13)(8,15,16,14)(17,19)(18,20)(21,28)(22,29)(23,24)(25,30)(26,31)(27,32);(1,5,7,4)(2,10,11,9)(3,13,6,12)(8,14,16,15)(17,20,23,31)(18,24,26,19)(21,27,22,30)(25,28,32,29)Polycyclic presentation:
2^9|a^2=e,b^2=hi,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 240249);