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 | |
3989 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)(3,9)(4,5)(6,8)(7,13)(10,15)(11,14)(12,16)(17,18)(19,24)(20,23)(21,25)(22,30)(26,32)(27,28)(29,31)(36,44)(39,46)(43,47)(45,48);(1,3,6,13)(2,7,8,9)(4,11,10,16)(5,12,15,14)(17,19)(18,24)(20,28)(21,30)(22,25)(23,27)(26,31)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,8)(4,10)(7,9)(11,16)(18,20)(19,27)(25,29)(26,30)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,4,6,10)(2,5,8,15)(3,11,13,16)(7,12,9,14)(17,20,23,18)(19,28,27,24)(21,29,31,25)(22,30,32,26)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,3)(2,9)(4,11)(5,14)(6,13)(7,8)(10,16)(12,15)(17,21,23,31)(18,25,20,29)(19,26,27,30)(22,24,32,28)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,5)(2,10)(3,12)(4,8)(6,15)(7,16)(9,11)(13,14)(17,22)(18,26)(19,25)(20,30)(21,28)(23,32)(24,31)(27,29)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=gh,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, 9620917);