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 |
| prime radical | #52599 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3998 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)(3,8,13,15)(4,5,10,9)(11,16,14,12)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34)(35,36)(37,42)(38,43)(39,40)(41,48)(44,46)(45,47);(2,7)(4,10)(8,15)(11,14)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(34,40)(35,41)(42,43)(44,45);(2,7)(5,9)(8,15)(12,16)(19,28)(22,31)(24,32)(26,29)(33,35,39,41)(34,36,40,48)(37,44,38,45)(42,46,43,47);(1,3,6,13)(2,8,7,15)(4,14,10,11)(5,12,9,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,41)(35,40)(37,46)(38,47)(39,48)(42,45)(43,44);(1,4,6,10)(2,9,7,5)(3,11,13,14)(8,12,15,16)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37,39,38)(34,42,40,43)(35,44,41,45)(36,46,48,47);(1,5,6,9)(2,10,7,4)(3,12,13,16)(8,11,15,14)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)(33,38,39,37)(34,43,40,42)(35,45,41,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=g,f^2=g,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, 8314291);