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 |
| prime radical | #39335 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,15)(12,13)(14,16)(18,23)(20,31)(21,33)(24,28)(29,39)(30,40)(38,45)(46,48);(2,7)(3,4)(10,13)(14,15)(18,23)(19,27)(20,31)(26,35)(30,40)(32,44)(36,41)(46,48);(1,3,6,4)(2,8,7,5)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,28,27,24)(20,25,31,34)(21,26,33,35)(29,41,39,36)(30,42,40,37)(32,38,44,45)(43,48,47,46);(1,4,6,3)(2,5,7,8)(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,36)(26,37)(31,39)(32,43)(33,40)(34,41)(35,42)(38,46)(44,47)(45,48);(1,5)(2,3)(4,7)(6,8)(17,20)(18,25)(19,29)(21,32)(22,31)(23,34)(24,36)(26,38)(27,39)(28,41)(30,43)(33,44)(35,45)(37,46)(40,47)(42,48);(17,21,22,33)(18,26,23,35)(19,30,27,40)(20,32,31,44)(24,37,28,42)(25,38,34,45)(29,43,39,47)(36,46,41,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022556);