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 | |
3855 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)(3,11)(4,6)(5,8)(7,10)(9,15)(12,16)(13,14)(17,18)(19,28)(20,24)(21,32)(22,27)(23,30)(25,31)(26,29)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,7,5)(2,8,10,11)(4,12,15,13)(6,14,9,16)(17,19)(18,23)(20,21)(22,30)(24,26)(25,32)(27,28)(29,31)(34,40)(35,37)(42,46)(43,45);(1,4)(2,9)(3,12)(5,13)(6,10)(7,15)(8,16)(11,14)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,26,28,32)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4)(2,9)(3,13)(5,12)(6,10)(7,15)(8,14)(11,16)(17,20)(18,25)(19,29)(21,30)(22,31)(23,26)(24,27)(28,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,5,7,3)(2,8,10,11)(4,13,15,12)(6,14,9,16)(17,18)(19,28)(20,24)(21,32)(22,27)(23,30)(25,31)(26,29)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,6,7,9)(2,4,10,15)(3,14,5,16)(8,12,11,13)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,24,28,25)(33,38,39,47)(34,42,40,46)(35,43,37,45)(36,44,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227526);