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 | |
4198 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)(3,8,11,15)(4,12,13,5)(9,10,14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,7)(4,13)(8,15)(9,14)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,35,39,44)(34,40,43,48)(36,47,41,38)(37,42,46,45);(2,7)(5,12)(8,15)(10,16)(18,22)(23,25)(24,28)(31,32)(35,44)(38,47)(40,48)(42,45);(1,3,6,11)(2,8,7,15)(4,14,13,9)(5,10,12,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,36)(34,37)(35,38)(39,41)(40,45)(42,48)(43,46)(44,47);(1,4)(2,5)(3,9)(6,13)(7,12)(8,16)(10,15)(11,14)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,37)(34,41)(35,45)(36,43)(38,48)(39,46)(40,47)(42,44);(1,5)(2,4)(3,10)(6,12)(7,13)(8,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,38)(34,42)(35,36)(37,48)(39,47)(40,46)(41,44)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8314424);