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 | |
4248 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18)(19,26)(20,30)(21,25)(22,23)(24,27)(28,29)(31,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,7)(4,8)(9,14)(12,15)(18,23)(19,27)(20,25)(29,32)(34,35)(39,41)(40,42)(46,47);(5,13)(9,14)(10,16)(12,15)(18,23)(21,30)(24,26)(29,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,26)(25,31)(30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20,22,25)(18,21,23,30)(19,28,27,31)(24,29,26,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21)(18,25)(19,29)(20,23)(22,30)(24,31)(26,28)(27,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=g,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=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8154849);