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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(9,14)(10,15)(11,16)(18,22)(19,25)(24,29)(26,31)(33,34)(35,41)(36,40)(37,45)(38,48)(39,42)(43,46)(44,47);(1,2)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,39,45)(34,37,42,41)(36,43,38,44)(40,47,48,46);(2,7)(3,8)(9,14)(10,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,38)(34,40,42,48)(35,43,45,44)(37,47,41,46);(1,3,6,8)(2,4,7,12)(5,10,13,15)(9,11,14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,39,41)(34,35,42,45)(36,46,38,47)(40,44,48,43);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35)(34,41)(36,43)(37,42)(38,44)(39,45)(40,46)(47,48);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,36)(34,40,42,48)(35,44,45,43)(37,47,41,46)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8019759);