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 | |
3983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,10,6)(5,9,15,12)(11,14,13,16)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28)(33,34)(35,41)(36,38)(37,43)(39,40)(42,46)(44,45)(47,48);(1,3,7,10)(2,8,4,6)(5,13,15,11)(9,16,12,14)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(34,40)(36,46)(41,43)(44,47);(2,4)(3,10)(9,12)(11,13)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,45,42,48);(1,4,7,2)(3,8,10,6)(5,9,15,12)(11,16,13,14)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,48)(43,45);(1,5)(2,9)(3,11)(4,12)(6,16)(7,15)(8,14)(10,13)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,45,42,48);(1,6,7,8)(2,10,4,3)(5,14,15,16)(9,11,12,13)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,44,43,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154615);