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 | |
3991 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)(3,8)(4,5)(6,7)(9,13)(10,16)(11,14)(12,15)(19,25)(23,26)(27,29)(31,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(2,7)(4,13)(8,15)(10,14)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,7)(5,9)(8,15)(11,16)(18,22)(23,26)(24,28)(31,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,14)(15,16)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,5,6,9)(2,4,7,13)(3,11,12,16)(8,14,15,10)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=i,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, 8323945);