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 | #52962 | C2 |
| 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(2,7)(4,13)(8,16)(10,14)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(34,40)(35,42)(41,44)(43,48);(2,7)(5,15)(8,16)(9,12)(19,28)(22,31)(24,32)(26,29)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,3)(2,8)(4,14)(5,9)(6,11)(7,16)(10,13)(12,15)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,15,16)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,5,6,15)(2,10,7,14)(3,9,11,12)(4,8,13,16)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314089);