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 | |
4064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,8)(4,13,14,5)(11,16,15,12)(17,18,23,21)(19,20,27,24)(22,25,30,29)(26,28,32,31)(33,34,39,43)(35,40,44,48)(36,37,41,47)(38,42,46,45);(2,7)(3,10)(4,14)(12,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,48)(46,47);(2,7)(5,13)(8,9)(12,16)(22,30)(25,29)(26,32)(28,31)(35,44)(38,46)(40,48)(42,45);(1,3,6,10)(2,8,7,9)(4,15,14,11)(5,12,13,16)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,36)(34,37)(35,45)(38,40)(39,41)(42,44)(43,47)(46,48);(1,4)(2,5)(3,11)(6,14)(7,13)(8,16)(9,12)(10,15)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,37)(34,41)(35,46)(36,43)(38,44)(39,47)(40,45)(42,48);(1,5)(2,4)(3,12)(6,13)(7,14)(8,15)(9,11)(10,16)(17,22,23,30)(18,25,21,29)(19,26,27,32)(20,28,24,31)(33,38,39,46)(34,42,43,45)(35,37,44,47)(36,40,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8281377);