Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | #61185 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1641 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(18,23)(19,27)(21,35)(25,37)(26,38)(28,43)(29,30)(31,42)(32,33)(36,40)(44,47)(45,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,23,22,26,35,38)(19,24,29,36,30,41,27,40)(20,31,33,42,34,37,32,25)(28,44,43,48,46,47,45,39);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,28)(21,29)(22,30)(23,36)(25,39)(26,41)(27,35)(31,44)(32,45)(33,43)(34,46)(37,47)(38,40)(42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,34)(18,25,26,42)(19,28,30,46)(21,32,35,33)(23,37,38,31)(24,39,41,48)(27,43,29,45)(36,47,40,44);(2,8)(6,9)(7,16)(14,15)(17,21,22,35)(18,23,26,38)(19,29,30,27)(20,32,34,33)(24,36,41,40)(25,37,42,31)(28,45,46,43)(39,47,48,44)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032386);