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 | |
4081 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)(4,9)(5,13)(6,7)(10,15)(11,14)(12,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(2,7)(4,12)(8,14)(10,13)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(34,40)(35,37)(42,46)(43,45);(2,7)(3,11)(9,16)(10,13)(18,20)(19,26)(25,28)(27,29)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,3,6,11)(2,8,7,14)(4,10,12,13)(5,9,15,16)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,4)(2,9)(3,10)(5,14)(6,12)(7,16)(8,15)(11,13)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,5,6,15)(2,10,7,13)(3,9,11,16)(4,8,12,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9633450);