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 | #45917 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,13,8)(4,12,14,9)(5,10,16,15)(18,22)(23,25)(24,29)(30,31)(33,34,39,41)(35,40,43,37)(36,42,45,47)(38,44,48,46);(2,7)(4,14)(8,11)(10,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,40)(36,38)(37,41)(39,43)(42,44)(45,48)(46,47);(2,7)(5,16)(8,11)(9,12)(20,28)(24,29)(26,32)(30,31)(34,41)(35,43)(38,48)(42,47);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,12,16,9)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36)(34,42)(35,38)(37,46)(39,45)(40,44)(41,47)(43,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,12)(8,16)(13,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,40)(34,43,41,35)(36,44,45,46)(38,47,48,42);(1,5,6,16)(2,10,7,15)(3,12,13,9)(4,11,14,8)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38)(34,44)(35,36)(37,47)(39,48)(40,42)(41,46)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8300960);