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 | |
4142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,9,13,7)(4,10,14,16)(5,12,15,11)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(35,41)(39,46)(42,44)(47,48);(1,3)(2,7)(4,12)(5,10)(6,13)(8,9)(11,14)(15,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,8)(5,15)(7,9)(10,16)(21,29)(25,28)(26,32)(30,31)(34,38)(39,46)(40,43)(47,48);(1,3)(2,9)(4,11)(5,10)(6,13)(7,8)(12,14)(15,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,4)(2,10)(3,12)(5,7)(6,14)(8,16)(9,15)(11,13)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,15)(2,11,8,12)(3,10,13,16)(4,7,14,9)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8832483);