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 | |
3981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,8,14,13)(10,16,12,15)(19,25)(23,26)(27,29)(31,32)(33,34)(35,36)(37,42)(38,43)(39,40)(41,48)(44,46)(45,47);(1,3,6,11)(2,7,5,9)(4,10,14,12)(8,15,13,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(34,40)(35,41)(42,43)(44,45);(2,5)(7,9)(8,13)(15,16)(18,22)(23,26)(24,28)(31,32)(33,35,39,41)(34,36,40,48)(37,44,38,45)(42,46,43,47);(1,4,6,14)(2,8,5,13)(3,10,11,12)(7,15,9,16)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,36)(34,41)(35,40)(37,46)(38,47)(39,48)(42,45)(43,44);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,14)(13,15)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,37,39,38)(34,42,40,43)(35,44,41,45)(36,46,48,47);(1,5,6,2)(3,9,11,7)(4,13,14,8)(10,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(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=h,c^2=gh,d^2=gi,e^2=i,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, 8323950);