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 | |
4005 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,22,25)(19,23,27,20)(21,24,30,32)(26,29,31,28)(33,34)(35,47)(36,42)(37,43)(38,44)(39,45)(40,46)(41,48);(4,11)(8,14)(10,15)(12,16)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,31,29)(33,35)(34,41)(36,43)(37,44)(38,39)(40,48)(42,45)(46,47);(2,7)(3,9)(8,14)(10,15)(17,19,22,27)(18,23,25,20)(21,26,30,31)(24,29,32,28)(33,36,40,38)(34,42,46,44)(35,43,48,39)(37,41,45,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37,40,45)(34,43,46,39)(35,44,48,42)(36,47,38,41);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,19)(21,29)(22,23)(24,31)(25,27)(26,32)(28,30)(33,38,40,36)(34,44,46,42)(35,43,48,39)(37,41,45,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,29)(25,32)(27,31)(33,39,40,43)(34,45,46,37)(35,38,48,36)(41,42,47,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9887542);