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:
(1,2,6,4)(3,9,11,7)(5,12,14,8)(10,15,13,16)(17,18,22,25)(19,26,20,23)(21,24,30,29)(27,32,28,31)(33,34,39,43)(35,38,37,41)(36,42,47,40)(44,46,45,48);(2,4)(7,9)(8,12)(15,16)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,35,39,37)(34,38,43,41)(36,45,47,44)(40,46,42,48);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,18,22,25)(19,23,20,26)(21,29,30,24)(27,32,28,31)(33,36,39,47)(34,40,43,42)(35,44,37,45)(38,48,41,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,37,39,35)(34,41,43,38)(36,44,47,45)(40,48,42,46);(1,4,6,2)(3,7,11,9)(5,12,14,8)(10,15,13,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,36)(34,42)(35,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,38,39,41)(34,35,43,37)(36,46,47,48)(40,45,42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=i,e^2=g,f^2=gh,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, 8900375);