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 | |
4144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(10,14)(11,16)(12,15)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34)(35,43)(36,39)(37,47)(38,48)(40,46)(41,45)(42,44);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(19,20)(23,26)(27,28)(31,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(2,7)(4,8)(10,14)(12,15)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,34,39,36)(35,40,46,43)(37,47,45,41)(38,42,44,48);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36,39,34)(35,40,46,43)(37,47,45,41)(38,48,44,42);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8028670);