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 | |
4137 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)(4,5)(6,7)(9,13)(10,16)(11,14)(12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,7)(4,13)(8,15)(10,14)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,35)(34,40)(36,38)(37,47)(39,44)(41,45)(42,46)(43,48);(2,7)(5,9)(8,15)(11,16)(20,28)(23,30)(26,32)(29,31)(35,44)(38,45)(40,48)(42,47);(1,3,6,12)(2,8,7,15)(4,14,13,10)(5,11,9,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36)(34,37)(35,45)(38,44)(39,41)(40,42)(43,46)(47,48);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,14)(15,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,37)(34,41)(35,42)(36,43)(38,40)(39,46)(44,47)(45,48);(1,5,6,9)(2,4,7,13)(3,11,12,16)(8,14,15,10)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38,39,45)(34,42,43,47)(35,36,44,41)(37,40,46,48)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325350);