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 | |
3936 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,28)(20,31)(21,29)(22,25)(23,27)(24,30)(26,32)(33,34,39,44)(35,40,37,42)(36,46,45,41)(38,43,47,48);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19)(18,24)(20,29)(21,25)(22,26)(23,30)(27,28)(31,32)(33,35,39,37)(34,40,44,42)(36,47,45,38)(41,43,46,48);(2,8)(4,9)(7,14)(10,15)(17,20,23,22)(18,25,27,31)(19,29,30,26)(21,28,32,24)(33,36,39,45)(34,41,44,46)(35,38,37,47)(40,48,42,43);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,21,23,32)(18,26,27,29)(19,25,30,31)(20,24,22,28)(33,37,39,35)(34,42,44,40)(36,38,45,47)(41,48,46,43);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,22,23,20)(18,25,27,31)(19,29,30,26)(21,24,32,28)(33,38)(34,43)(35,45)(36,37)(39,47)(40,41)(42,46)(44,48);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,19)(18,24)(20,29)(21,25)(22,26)(23,30)(27,28)(31,32)(33,34)(35,40)(36,41)(37,42)(38,48)(39,44)(43,47)(45,46)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013877);