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 | |
4173 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,23,26,20)(21,29,31,25)(27,28,32,30)(33,34,39,38)(35,43,46,40)(36,41,45,47)(37,42,44,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,26)(18,23,24,20)(21,27,31,32)(25,30,29,28)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,42)(47,48);(3,10)(6,7)(11,16)(14,15)(18,24)(19,26)(25,29)(27,32)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,42,43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,23)(20,26)(21,25)(22,24)(27,30)(28,32)(29,31)(33,36)(34,41)(35,44)(37,46)(38,47)(39,45)(40,42)(43,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,20,22,23)(18,19,24,26)(21,30,31,28)(25,32,29,27)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,41,43,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,30)(24,29)(26,32)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,42,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,e^2=gh,f^2=g,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, 9074879);