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 | |
4381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(2,6)(7,10)(8,12)(14,16)(18,20)(24,25)(26,28)(31,32)(36,45)(40,44)(42,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37,38,34)(35,39,43,41)(36,44,45,40)(42,46,48,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48)Polycyclic presentation:
2^9|b^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659167);