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 | |
3907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,11,4)(5,8,13,14)(10,15,16,12)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(2,7)(4,9)(8,14)(12,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,34)(35,40)(36,37)(38,39)(41,46)(42,48)(43,45)(44,47);(5,13)(8,14)(10,16)(12,15)(18,21)(23,25)(24,28)(31,32)(34,39)(37,41)(40,47)(43,48);(1,3,6,11)(2,4,7,9)(5,10,13,16)(8,12,14,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,35)(34,40)(36,45)(37,43)(38,44)(39,47)(41,48)(42,46);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,36)(34,41)(35,42)(37,39)(38,46)(40,43)(44,45)(47,48);(1,5,6,13)(2,8,7,14)(3,10,11,16)(4,12,9,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,37,38,41)(34,36,39,46)(35,43,44,48)(40,45,47,42)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279336);