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 | #44122 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,6,11,16)(4,9,14,15)(5,13,12,10)(34,39)(35,42)(36,44)(46,47);(1,3,7,11)(2,6,8,16)(4,5,14,12)(9,13,15,10)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,34)(35,37)(36,45)(38,39)(40,44)(41,42)(43,46)(47,48);(2,8)(3,11)(5,12)(9,15)(18,23)(20,29)(24,31)(26,32)(33,35,38,42)(34,37,39,41)(36,43,44,48)(40,47,45,46);(1,4)(2,9)(3,12)(5,11)(6,10)(7,14)(8,15)(13,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,36)(34,40)(35,43)(37,47)(38,44)(39,45)(41,46)(42,48);(1,5,7,12)(2,10,8,13)(3,4,11,14)(6,15,16,9)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,35)(34,41)(36,43)(37,39)(38,42)(40,46)(44,48)(45,47);(1,6)(2,11)(3,8)(4,13)(5,15)(7,16)(9,12)(10,14)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,37,38,41)(34,35,39,42)(36,46,44,47)(40,48,45,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=gh,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, 8541051);