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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4247 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,16)(14,15)(33,34)(35,44)(36,47)(37,38)(39,40)(41,45)(42,46)(43,48);(2,7)(5,8)(9,15)(13,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(34,40)(35,45)(37,43)(42,47);(1,3,6,11)(2,5,7,8)(4,12,14,10)(9,16,15,13)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35)(34,41)(36,43)(37,46)(38,47)(39,45)(40,44)(42,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,16)(12,14)(13,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36,39,46)(34,42,40,47)(35,43,45,37)(38,41,48,44);(1,4)(2,9)(3,10)(5,13)(6,14)(7,15)(8,16)(11,12)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,39,43)(34,38,40,48)(35,46,45,36)(41,47,44,42);(1,5,6,8)(2,3,7,11)(4,13,14,16)(9,10,15,12)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,38)(34,43)(35,47)(36,41)(37,40)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8729149);