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 | #2424 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3960 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)(3,8)(4,13)(5,9)(6,7)(10,11)(12,15)(14,16)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(2,7)(4,9)(8,15)(10,14)(17,19,23,28)(18,24,27,32)(20,31,25,22)(21,26,30,29)(34,39)(35,41)(40,46)(42,47);(2,7)(5,13)(8,15)(11,16)(19,28)(22,31)(24,32)(26,29)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,3,6,12)(2,8,7,15)(4,14,9,10)(5,11,13,16)(17,20)(18,21)(19,22)(23,25)(24,29)(26,32)(27,30)(28,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,4,6,9)(2,5,7,13)(3,10,12,14)(8,16,15,11)(17,21)(18,25)(19,29)(20,27)(22,32)(23,30)(24,31)(26,28)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48);(1,5)(2,9)(3,11)(4,7)(6,13)(8,10)(12,16)(14,15)(17,22)(18,26)(19,20)(21,32)(23,31)(24,30)(25,28)(27,29)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,d^2=h,e^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, 8314049);