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 | |
3956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,2,6,7)(3,8,13,16)(4,9,14,12)(5,10,15,11)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,7)(4,14)(8,16)(10,11)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,3)(2,8)(4,11)(5,9)(6,13)(7,16)(10,14)(12,15)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,4)(2,9)(3,11)(5,8)(6,14)(7,12)(10,13)(15,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,5)(2,10)(3,12)(4,8)(6,15)(7,11)(9,13)(14,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424702);