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 | #52632 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3986 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)(3,5)(4,8)(6,10)(7,9)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,25)(22,31)(23,26)(27,28)(29,32)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,3,6,9)(2,7,10,5)(4,11,14,16)(8,15,12,13)(17,19,23,29)(18,24,26,32)(20,25,31,27)(21,22,28,30)(34,39)(36,40)(41,45)(44,47);(1,2,6,10)(3,7,9,5)(4,12,14,8)(11,13,16,15)(17,20,23,31)(18,22,26,30)(19,27,29,25)(21,32,28,24)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,2)(3,7)(4,12)(5,9)(6,10)(8,14)(11,13)(15,16)(17,20)(18,22)(19,25)(21,32)(23,31)(24,28)(26,30)(27,29)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,21,23,28)(18,25,26,27)(19,22,29,30)(20,24,31,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5,6,7)(2,9,10,3)(4,13,14,15)(8,16,12,11)(17,22)(18,20)(19,28)(21,29)(23,30)(24,25)(26,31)(27,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|b^2=gi,c^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961127);