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 | |
4189 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18)(19,20)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(2,7)(3,11)(4,14)(13,15)(17,19,23,25)(18,20,21,28)(22,26,31,32)(24,27,29,30)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(2,7)(4,14)(8,10)(12,16)(18,21)(19,25)(24,29)(26,32)(36,45)(40,48)(43,44)(46,47);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,20)(18,19)(21,25)(22,30)(23,28)(24,32)(26,29)(27,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21,23,18)(19,20,25,28)(22,29,31,24)(26,27,32,30)(33,36,38,45)(34,40,41,48)(35,43,37,44)(39,46,42,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22)(18,24)(19,26)(20,27)(21,29)(23,31)(25,32)(28,30)(33,37,38,35)(34,39,41,42)(36,44,45,43)(40,46,48,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495582);