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 | |
3846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,2,6,7)(3,5,9,14)(4,8,13,15)(10,12,11,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,7)(5,14)(8,15)(12,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3,6,9)(2,5,7,14)(4,11,13,10)(8,16,15,12)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4)(2,8)(3,10)(5,12)(6,13)(7,15)(9,11)(14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,11)(10,15)(13,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202766);