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 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,13,15)(4,5,10,9)(11,16,14,12)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,7)(4,10)(8,15)(11,14)(17,19,22,24)(18,23,26,20)(21,27,29,31)(25,30,32,28)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(2,7)(5,9)(8,15)(12,16)(21,29)(25,32)(27,31)(28,30)(34,38)(39,41)(40,44)(47,48);(1,3)(2,8)(4,14)(5,12)(6,13)(7,15)(9,16)(10,11)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4,6,10)(2,9,7,5)(3,11,13,14)(8,12,15,16)(17,20)(18,24)(19,26)(21,28)(22,23)(25,31)(27,32)(29,30)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,5,6,9)(2,10,7,4)(3,12,13,16)(8,11,15,14)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314432);