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 | |
4495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(9,10,13,12)(11,15,16,14)(17,18,23,28)(19,25,20,24)(21,26,35,33)(22,36,38,27)(29,41,31,39)(30,40,32,42)(34,44,37,43)(45,47,46,48);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19,23,20)(18,24,28,25)(21,31,35,29)(22,30,38,32)(26,41,33,39)(27,40,36,42)(34,46,37,45)(43,48,44,47);(1,3)(2,5)(4,8)(6,7)(17,20,23,19)(18,25,28,24)(21,31,35,29)(22,32,38,30)(26,41,33,39)(27,42,36,40)(34,46,37,45)(43,48,44,47);(2,6)(5,7)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,33)(22,27)(23,28)(26,35)(29,39)(30,42)(31,41)(32,40)(34,44)(36,38)(37,43)(45,47)(46,48);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,21)(18,26)(19,29)(20,31)(22,37)(23,35)(24,39)(25,41)(27,44)(28,33)(30,46)(32,45)(34,38)(36,43)(40,48)(42,47);(1,3)(2,7)(4,8)(5,6)(9,12,13,10)(11,15,16,14)(17,22,23,38)(18,27,28,36)(19,30,20,32)(21,34,35,37)(24,40,25,42)(26,43,33,44)(29,45,31,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gh,f^2=g,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, 10125829);