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 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,9,13,16)(11,12,15,14)(17,18)(19,24)(20,26)(21,22)(23,28)(25,29)(27,32)(30,31);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,23,30)(18,24,28,31)(20,27,29,21)(22,26,32,25)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(3,8)(4,7)(5,13)(9,16)(17,20,23,29)(18,25,28,26)(19,21,30,27)(22,24,32,31)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,3)(2,7)(4,10)(5,11)(6,8)(9,14)(12,16)(13,15)(17,20)(18,26)(19,27)(21,30)(22,31)(23,29)(24,32)(25,28)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,4,6,7)(2,8,10,3)(5,12,13,14)(9,15,16,11)(17,21,23,27)(18,22,28,32)(19,29,30,20)(24,25,31,26)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,27)(19,26)(20,31)(21,28)(23,32)(24,29)(25,30)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901821);