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 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,10)(5,12,13,8)(11,16,15,14)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(34,40)(35,38)(36,46)(45,48);(3,9)(7,10)(11,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,34)(35,43)(36,37)(38,41)(39,40)(42,46)(44,45)(47,48);(2,4)(7,10)(8,12)(14,16)(19,25)(22,29)(26,31)(30,32)(33,35,39,38)(34,41,40,43)(36,44,46,47)(37,48,42,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,42)(35,44)(37,40)(38,47)(39,46)(41,45)(43,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37,39,42)(34,36,40,46)(35,45,38,48)(41,47,43,44);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,38,39,35)(34,41,40,43)(36,47,46,44)(37,48,42,45)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,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002272);