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 | |
3937 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,30)(31,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(2,7)(4,8)(10,13)(12,14)(17,19,23,30)(18,24,28,29)(20,21,27,31)(22,26,25,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,7)(3,9)(10,13)(11,16)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,20,23,27)(18,25,28,22)(19,21,30,31)(24,32,29,26)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,21)(18,26)(19,27)(20,30)(22,29)(23,31)(24,25)(28,32)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,22,23,25)(18,27,28,20)(19,26,30,32)(21,24,31,29)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=g,e^2=hi,f^2=gi,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, 9636409);