Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61283 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,14,8)(11,15,13,16)(17,18)(19,30)(20,35)(21,27)(22,28)(23,41)(24,29)(25,32)(26,36)(31,45)(33,44)(34,43)(37,42)(38,46)(39,47)(40,48);(1,3,5,10)(2,6,9,7)(4,13,14,11)(8,16,12,15)(17,19)(18,25)(20,21)(22,32)(23,33)(24,34)(26,27)(28,43)(29,44)(30,41)(31,38)(35,46)(36,39)(37,40)(42,47)(45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,16)(12,15)(13,14)(17,20,22,31,24,37,23,36)(18,26,41,42,29,45,28,35)(19,21,32,38,34,40,33,39)(25,27,30,47,44,48,43,46);(3,10)(6,7)(11,13)(15,16)(17,21)(18,27)(19,31)(20,33)(22,38)(23,39)(24,40)(25,42)(26,43)(28,46)(29,48)(30,45)(32,37)(34,36)(35,44)(41,47);(1,4,5,14)(2,8,9,12)(3,11,10,13)(6,15,7,16)(17,22,24,23)(18,28,29,41)(19,32,34,33)(20,31,37,36)(21,38,40,39)(25,43,44,30)(26,35,45,42)(27,46,48,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7100381);