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 | |
4362 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,25)(20,24)(21,27)(22,23)(26,31)(28,30)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(3,11)(4,12)(6,16)(8,13)(18,23)(21,25)(24,28)(29,31)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,18)(19,21)(20,28)(22,23)(24,30)(25,27)(26,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,48)(45,46);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834278);