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 | |
4117 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)(17,18,23,25)(19,27,28,24)(20,22,29,30)(21,26,31,32)(33,34)(35,41)(36,47)(37,40)(38,42)(39,44)(43,46)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,28)(18,24,25,27)(20,26,29,32)(21,30,31,22)(33,35,40,48)(34,41,37,45)(36,43,42,44)(38,39,47,46);(3,10)(6,7)(11,16)(14,15)(18,25)(20,29)(24,27)(26,32)(33,36,40,42)(34,38,37,47)(35,44,48,43)(39,41,46,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,22)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,37,40,34)(35,45,48,41)(36,38,42,47)(39,44,46,43);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,23,31)(18,26,25,32)(19,22,28,30)(20,27,29,24)(33,38)(34,42)(35,46)(36,37)(39,48)(40,47)(41,43)(44,45);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,22)(18,20)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,39,40,46)(34,43,37,44)(35,47,48,38)(36,41,42,45)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gh,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, 9076540);