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 | |
4137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,34)(35,39)(36,41)(37,45)(38,40)(42,48)(43,44)(46,47);(2,3)(4,11)(8,10)(12,16)(18,23)(20,30)(24,26)(27,31)(33,35,38,41)(34,39,40,36)(37,46,48,43)(42,44,45,47);(5,13)(8,10)(12,16)(14,15)(18,23)(21,25)(24,26)(28,32)(34,40)(35,41)(42,45)(43,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,29)(18,24,23,26)(20,31,30,27)(21,28,25,32)(33,36,38,39)(34,41,40,35)(37,47,48,44)(42,43,45,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,32)(29,31)(33,37,38,48)(34,42,40,45)(35,43,41,46)(36,44,39,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,25)(18,20,23,30)(19,28,29,32)(24,31,26,27)(33,34)(35,39)(36,41)(37,42)(38,40)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300567);