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 | |
4210 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,21,23)(19,24,27,22)(20,28,29,25)(26,30,32,31)(33,34)(35,41)(36,45)(37,42)(38,43)(39,46)(40,47)(44,48);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,19,21,27)(18,22,23,24)(20,26,29,32)(25,30,28,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,43)(42,47);(2,9)(3,4)(6,15)(13,16)(18,23)(20,29)(22,24)(26,32)(36,47)(37,48)(40,45)(42,44);(1,4,7,3)(2,5,9,8)(6,12,15,11)(10,13,14,16)(17,18)(19,22)(20,25)(21,23)(24,27)(26,30)(28,29)(31,32)(33,36,38,47)(34,40,43,45)(35,44,46,42)(37,39,48,41);(1,5)(2,4)(3,9)(6,13)(7,8)(10,12)(11,14)(15,16)(17,19)(18,24)(20,26)(21,27)(22,23)(25,31)(28,30)(29,32)(33,35)(34,41)(36,42)(37,45)(38,46)(39,43)(40,48)(44,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,20,21,29)(18,25,23,28)(19,26,27,32)(22,30,24,31)(33,37,38,48)(34,42,43,44)(35,45,46,40)(36,39,47,41)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812360);