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 | |
4286 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)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,34,38,42)(35,39,45,40)(36,37,41,47)(43,48,46,44);(2,7)(3,10)(8,15)(11,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,42)(41,43)(47,48);(2,7)(4,9)(8,15)(12,16)(19,25)(22,29)(26,31)(30,32)(33,35,38,45)(34,40,42,39)(36,43,41,46)(37,44,47,48);(1,3)(2,4)(5,13)(6,10)(7,9)(8,16)(11,14)(12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,36)(34,37)(35,43)(38,41)(39,48)(40,44)(42,47)(45,46);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,42)(41,46)(47,48);(1,5,6,14)(2,8,7,15)(3,11,10,13)(4,12,9,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37)(34,41)(35,44)(36,42)(38,47)(39,43)(40,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,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, 8441990);