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 | |
4068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,4,11,8)(6,13,14,9)(10,15,16,12)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,38,43)(35,39,37,41)(36,42,46,40)(44,48,45,47);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,38,37)(34,39,43,41)(36,45,46,44)(40,48,42,47);(2,5)(4,8)(6,14)(10,16)(19,26)(20,24)(21,29)(23,30)(33,36,38,46)(34,40,43,42)(35,44,37,45)(39,47,41,48);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,37,38,35)(34,41,43,39)(36,44,46,45)(40,47,42,48);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,36)(34,42)(35,44)(37,45)(38,46)(39,48)(40,43)(41,47);(1,6,7,14)(2,9,5,13)(3,10,11,16)(4,12,8,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,34)(35,39)(36,42)(37,41)(38,43)(40,46)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8880321);