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 | |
3979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(19,25)(23,26)(27,29)(31,32)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(34,40)(35,36)(42,47)(43,44);(4,12)(7,14)(10,16)(13,15)(18,22)(23,26)(24,28)(31,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=gi,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, 8323956);