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 | |
3857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,11)(4,10,15,14)(6,13,16,12)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3)(2,8)(4,13)(5,7)(6,14)(9,11)(10,16)(12,15)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,9)(3,5)(4,15)(6,16)(18,23)(19,27)(20,30)(21,31)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8886950);