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 | #48239 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4145 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)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(34,39)(36,40)(41,47)(44,45);(1,3,7,9)(2,4,5,12)(6,10,15,16)(8,11,13,14)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,45,48,44);(2,5)(3,9)(8,13)(10,16)(18,23)(20,29)(24,31)(26,32)(33,35,38,43)(34,40,39,36)(37,42,46,48)(41,45,47,44);(1,4)(2,3)(5,9)(6,14)(7,12)(8,16)(10,13)(11,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36)(34,35)(37,45)(38,40)(39,43)(41,48)(42,47)(44,46);(1,5,7,2)(3,4,9,12)(6,13,15,8)(10,11,16,14)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,47)(46,48);(1,6)(2,8)(3,10)(4,11)(5,13)(7,15)(9,16)(12,14)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,45)(43,48)Polycyclic presentation:
2^9|b^2=gh,c^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541003);