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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,12)(10,16)(14,15)(17,18,22,23)(19,26,21,24)(20,28,29,25)(27,31,30,32)(33,34,39,40)(35,37,42,47)(36,43,38,41)(44,48,45,46);(2,4)(7,9)(8,12)(14,15)(18,23)(20,29)(24,26)(27,30)(34,40)(37,47)(41,43)(46,48);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,19,22,21)(18,24,23,26)(20,27,29,30)(25,31,28,32)(33,35,39,42)(34,37,40,47)(36,44,38,45)(41,46,43,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,36,39,38)(34,41,40,43)(35,44,42,45)(37,46,47,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,29)(18,25,23,28)(19,27,21,30)(24,31,26,32)(33,37)(34,42)(35,40)(36,46)(38,48)(39,47)(41,45)(43,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,19)(18,26,23,24)(20,30,29,27)(25,32,28,31)(33,38,39,36)(34,43,40,41)(35,45,42,44)(37,48,47,46)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=ghi,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^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017715);