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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,46)(47,48);(3,9)(6,13)(10,15)(14,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,35,39,46)(34,40,44,45)(36,37,43,47)(38,42,41,48);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,36,39,43)(34,41,44,38)(35,37,46,47)(40,48,45,42);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,35)(34,40)(36,37)(38,42)(39,46)(41,48)(43,47)(44,45);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,37)(34,42)(35,43)(36,46)(38,45)(39,47)(40,41)(44,48);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,38,39,41)(34,43,44,36)(35,42,46,48)(37,40,47,45)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,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, 10154134);