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 | |
4227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18,23,21)(19,24,20,25)(22,26,30,29)(27,31,28,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(18,21)(24,25)(26,29)(31,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,9)(3,4)(10,16)(11,12)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,23,18)(19,25,20,24)(22,26,30,29)(27,31,28,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=g,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, 9817399);