Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #56292 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,4,11,7)(5,13,14,9)(10,15,16,12)(18,21)(19,25)(24,29)(26,31)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,6,11)(2,7,8,4)(5,10,14,16)(9,15,13,12)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(34,39)(35,42)(41,44)(43,48);(2,8)(4,7)(9,13)(12,15)(19,25)(20,28)(23,27)(24,29)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,2)(3,4)(5,9)(6,8)(7,11)(10,12)(13,14)(15,16)(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,6,14)(2,9,8,13)(3,10,11,16)(4,12,7,15)(17,21,22,18)(19,27,25,23)(20,29,28,24)(26,32,31,30)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034698);