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 | |
3999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,8,14,13)(10,16,12,15)(18,22)(19,25)(24,29)(26,31)(33,34,40,44)(35,41,38,39)(36,43,37,42)(45,48,46,47);(1,3,6,11)(2,7,5,9)(4,10,14,12)(8,15,13,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,41)(36,46)(37,45)(38,40)(39,44)(42,48)(43,47);(2,5)(7,9)(8,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,40,37)(34,42,44,43)(35,45,38,46)(39,48,41,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,40,36)(34,43,44,42)(35,45,38,46)(39,48,41,47);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,14)(13,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,38,40,35)(34,41,44,39)(36,46,37,45)(42,47,43,48);(1,5,6,2)(3,9,11,7)(4,13,14,8)(10,16,12,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,39,40,41)(34,35,44,38)(36,47,37,48)(42,46,43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8313692);