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 | |
4169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(1,3,7,4)(2,9,5,8)(6,11,14,12)(10,16,13,15)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36)(34,41)(35,44)(37,43)(38,45)(39,46)(40,47)(42,48);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,37,39,41)(34,36,43,46)(35,42,45,47)(38,40,44,48);(1,5,7,2)(3,8,4,9)(6,13,14,10)(11,15,12,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10349719);