Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18)(19,31)(20,37)(21,27)(22,30)(23,29)(24,28)(25,35)(26,39)(32,44)(33,47)(34,42)(36,43)(38,41)(40,45)(46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,22,35,24,34,23,36)(18,25,29,31,28,43,30,42)(20,38,39,33,21,32,40,48)(26,44,37,47,45,41,27,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,26)(19,32)(21,24)(22,39)(23,40)(25,41)(27,30)(28,45)(29,37)(31,46)(33,36)(34,38)(35,48)(42,47)(43,44);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21,24,20)(18,27,28,37)(19,33,34,48)(22,39,23,40)(25,41,43,44)(26,29,45,30)(31,47,42,46)(32,35,38,36);(2,8)(6,9)(7,16)(14,15)(18,28)(19,34)(26,45)(27,37)(29,30)(32,38)(33,48)(35,36)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=h,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,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, 7032365);