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 | |
4538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(9,10,13,14)(11,12,15,16)(18,19)(20,30)(22,36)(25,28)(31,33)(34,46)(39,42)(47,48);(1,2)(3,4)(5,6)(7,8)(10,14)(11,15)(17,18,23,19)(20,24,30,27)(21,28,35,25)(22,26,36,29)(31,40,33,38)(32,39,37,42)(34,43,46,41)(44,48,45,47);(2,6)(3,7)(9,11)(10,12)(13,15)(14,16)(17,19,23,18)(20,24,30,27)(21,25,35,28)(22,29,36,26)(31,40,33,38)(32,39,37,42)(34,41,46,43)(44,48,45,47);(3,7)(4,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,24)(19,27)(21,33)(22,37)(23,30)(25,40)(26,42)(28,38)(29,39)(31,35)(32,36)(34,44)(41,47)(43,48)(45,46);(1,3)(2,4)(5,7)(6,8)(17,21,23,35)(18,25,19,28)(20,31,30,33)(22,34,36,46)(24,38,27,40)(26,41,29,43)(32,44,37,45)(39,47,42,48);(1,4)(2,3)(5,8)(6,7)(17,22)(18,26)(19,29)(20,32)(21,34)(23,36)(24,39)(25,41)(27,42)(28,43)(30,37)(31,44)(33,45)(35,46)(38,47)(40,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532607);