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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(3,9)(6,12)(10,15)(13,16)(18,24)(20,25)(26,30)(28,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,30,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065925);