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 | |
4039 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)(3,8)(4,12)(5,7)(6,11)(9,14)(10,15)(13,16)(17,18)(19,23)(20,21)(22,25)(24,28)(26,27)(29,32)(30,31)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(35,37)(39,42)(43,45)(47,48);(2,7)(6,8)(9,12)(15,16)(17,19,22,28)(18,23,25,24)(20,26,30,29)(21,27,31,32)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,25)(26,30)(31,32)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,20,22,30)(18,21,25,31)(19,26,28,29)(23,27,24,32)(33,37,38,35)(34,39,41,42)(36,43,46,45)(40,48,44,47);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,21)(18,20)(19,27)(22,31)(23,26)(24,29)(25,30)(28,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113967);