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 | |
4299 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)(3,9)(4,10)(5,8)(6,7)(11,16)(12,13)(14,15)(18,22)(23,29)(24,27)(30,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(2,7)(3,10)(8,15)(11,13)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,7)(4,9)(8,15)(12,16)(19,25)(23,29)(26,31)(30,32)(33,35,37,43)(34,39,41,38)(36,42,45,48)(40,47,44,46);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,34)(35,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|c^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441925);