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 | #42591 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,5,8,10)(4,7,15,13)(11,14,12,16)(18,22)(20,27)(23,28)(26,31)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3)(2,5)(4,12)(6,8)(7,16)(9,10)(11,15)(13,14)(19,25)(23,28)(26,31)(30,32)(34,38)(35,42)(40,44)(43,48);(3,8)(5,10)(11,12)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,2)(3,10)(4,13)(5,8)(6,9)(7,15)(11,14)(12,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,4)(2,7)(3,11)(5,14)(6,15)(8,12)(9,13)(10,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,5,6,10)(2,8,9,3)(4,14,15,16)(7,12,13,11)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8325793);