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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3844 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)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,9)(3,4)(6,15)(13,16)(18,23)(19,24)(21,29)(27,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5)(2,4)(3,9)(6,13)(7,8)(10,12)(11,14)(15,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815653);