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 | |
4056 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)(3,10)(4,9)(5,14)(6,8)(7,12)(11,16)(13,15)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(34,38)(36,44)(39,41)(42,48);(1,3)(2,7)(4,11)(5,9)(6,12)(8,10)(13,14)(15,16)(18,21)(24,25)(26,29)(31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(2,8)(4,13)(7,10)(11,14)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,34,37,38)(35,39,43,41)(36,40,44,45)(42,46,48,47);(1,4,6,13)(2,9,8,15)(3,11,12,14)(5,10,16,7)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,10)(4,11)(5,15)(6,12)(7,8)(9,16)(13,14)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,16)(2,11,8,14)(3,9,12,15)(4,10,13,7)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9831204);