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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,7)(5,8)(10,13)(11,15)(17,18,23,24)(19,27,28,20)(21,25,33,37)(22,35,36,26)(29,39,41,31)(30,32,42,40)(34,46,45,38)(43,44,48,47);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,28)(21,33)(26,35)(30,42)(31,39)(34,45)(44,47);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19,23,28)(18,20,24,27)(21,29,33,41)(22,30,36,42)(25,31,37,39)(26,32,35,40)(34,43,45,48)(38,44,46,47);(17,20)(18,19)(21,31)(22,32)(23,27)(24,28)(25,29)(26,30)(33,39)(34,44)(35,42)(36,40)(37,41)(38,43)(45,47)(46,48);(17,21)(18,25)(19,29)(20,31)(22,34)(23,33)(24,37)(26,38)(27,39)(28,41)(30,43)(32,44)(35,46)(36,45)(40,47)(42,48);(17,22,23,36)(18,26,24,35)(19,30,28,42)(20,32,27,40)(21,34,33,45)(25,38,37,46)(29,43,41,48)(31,44,39,47)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537856);