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 | #40768 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(17,18)(19,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32);(2,6)(3,10)(8,14)(11,12)(17,19,23,25)(18,20,21,28)(22,26,31,32)(24,27,29,30)(33,34)(35,40)(36,37)(38,39)(41,45)(42,46)(43,48)(44,47);(2,6)(7,9)(8,14)(15,16)(18,21)(19,25)(24,29)(26,32)(34,39)(36,45)(40,47)(42,48);(1,3,5,10)(2,7,6,9)(4,12,13,11)(8,16,14,15)(17,20)(18,19)(21,25)(22,30)(23,28)(24,32)(26,29)(27,31)(33,35)(34,40)(36,42)(37,46)(38,44)(39,47)(41,43)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21,23,18)(19,20,25,28)(22,29,31,24)(26,27,32,30)(33,36)(34,41)(35,42)(37,39)(38,45)(40,43)(44,48)(46,47);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,22)(18,24)(19,26)(20,27)(21,29)(23,31)(25,32)(28,30)(33,37,38,41)(34,36,39,45)(35,43,44,46)(40,48,47,42)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^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, 8541024);