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 | #2615 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3930 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(2,7)(3,10)(9,12)(11,13)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(34,39)(35,42)(41,44)(43,48);(2,7)(5,8)(9,12)(14,16)(18,23)(21,30)(24,31)(28,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202848);