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 | |
4054 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)(3,10,8,4)(5,13,14,9)(11,12,15,16)(17,18,22,26)(19,25,21,23)(20,24,29,32)(27,31,28,30)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(2,7)(3,8)(9,13)(11,15)(17,19,22,21)(18,23,26,25)(20,27,29,28)(24,30,32,31)(34,39)(35,42)(41,44)(43,48);(2,7)(4,10)(9,13)(12,16)(17,20,22,29)(18,24,26,32)(19,27,21,28)(23,30,25,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3,6,8)(2,4,7,10)(5,11,14,15)(9,12,13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,6,14)(2,9,7,13)(3,11,8,15)(4,12,10,16)(17,21,22,19)(18,25,26,23)(20,28,29,27)(24,31,32,30)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026846);