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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,8)(10,13)(14,16)(17,18,22,23)(19,24,29,35)(20,25,32,36)(21,33,34,26)(27,37,30,39)(28,42,43,38)(31,45,46,40)(41,48,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,23)(24,35)(25,36)(26,33)(37,39)(38,42)(40,45)(47,48);(1,2)(3,7)(4,8)(5,6)(17,18)(19,24)(20,25)(21,26)(22,23)(27,37)(28,38)(29,35)(30,39)(31,40)(32,36)(33,34)(41,47)(42,43)(44,48)(45,46);(1,3,5,4)(2,7,6,8)(17,19,22,29)(18,24,23,35)(20,30,32,27)(21,28,34,43)(25,39,36,37)(26,38,33,42)(31,44,46,41)(40,48,45,47);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,25,23,36)(19,27,29,30)(21,31,34,46)(24,37,35,39)(26,40,33,45)(28,41,43,44)(38,47,42,48);(17,21,22,34)(18,26,23,33)(19,28,29,43)(20,31,32,46)(24,38,35,42)(25,40,36,45)(27,41,30,44)(37,47,39,48)Polycyclic presentation:
2^9|a^2=i,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538132);