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 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,26)(19,23,24,20)(21,25,31,32)(27,29,30,28)(33,34)(35,44)(36,48)(37,46)(38,42)(39,43)(40,47)(41,45);(1,2,6,7)(3,5,9,14)(4,8,12,15)(10,13,16,11)(17,19)(18,23)(20,26)(21,27)(22,24)(25,29)(28,32)(30,31)(33,35)(34,40)(36,37)(38,46)(39,47)(41,48)(42,45)(43,44);(2,7)(3,9)(8,15)(10,16)(17,20,22,23)(18,24,26,19)(21,28,31,29)(25,30,32,27)(33,36,39,42)(34,38,43,48)(35,37,47,45)(40,46,44,41);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,15)(12,16)(17,20)(18,19)(21,29)(22,23)(24,26)(25,30)(27,32)(28,31)(33,36)(34,38)(35,45)(37,47)(39,42)(40,41)(43,48)(44,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,19)(18,20)(21,30)(22,24)(23,26)(25,29)(27,31)(28,32)(33,37,39,45)(34,41,43,46)(35,36,47,42)(38,40,48,44);(1,5)(2,9)(3,7)(4,11)(6,14)(8,10)(12,13)(15,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,30)(26,32)(33,38)(34,42)(35,46)(36,43)(37,44)(39,48)(40,45)(41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616481);