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 | |
4069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18)(19,28)(20,22)(21,25)(23,27)(24,29)(26,31)(30,32)(33,34)(35,46)(36,39)(37,47)(38,43)(40,45)(41,44)(42,48);(4,11)(7,9)(12,16)(14,15)(19,29)(21,30)(24,28)(25,32)(33,35)(34,38)(36,47)(37,41)(39,48)(40,43)(42,44)(45,46);(2,3)(7,9)(8,10)(14,15)(17,19,23,29)(18,24,27,28)(20,30,26,21)(22,25,31,32)(33,36,40,44)(34,41,45,39)(35,47,43,42)(37,46,48,38);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20,23,26)(18,22,27,31)(19,21,29,30)(24,32,28,25)(33,37,40,48)(34,42,45,47)(35,41,43,39)(36,38,44,46);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21)(18,25)(19,20)(22,28)(23,30)(24,31)(26,29)(27,32)(33,38)(34,43)(35,45)(36,37)(39,47)(40,46)(41,42)(44,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,39,40,41)(34,44,45,36)(35,48,43,37)(38,42,46,47)Polycyclic presentation:
2^9|a^2=ghi,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013834);