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 | |
3927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,4,7)(5,13,14,10)(11,15,12,16)(20,28)(23,30)(26,32)(29,31)(33,34,39,42)(35,43,46,40)(36,47,45,37)(38,48,44,41);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,46)(42,43)(47,48);(2,8)(7,9)(10,13)(15,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,39,42)(35,40,46,43)(36,47,45,37)(38,41,44,48);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,39,45)(34,37,42,47)(35,44,46,38)(40,41,43,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,37)(34,36)(35,41)(38,43)(39,47)(40,44)(42,45)(46,48);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,38,39,44)(34,41,42,48)(35,45,46,36)(37,43,47,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8079816);