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 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(5,13)(7,9)(12,16)(21,29)(24,28)(26,32)(30,31)(33,34,39,40)(35,36,46,41)(37,42,38,43)(44,47,45,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(34,40)(36,41)(42,43)(47,48);(4,11)(7,9)(12,16)(14,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35)(34,41)(36,40)(37,44)(38,45)(39,46)(42,48)(43,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(33,36,39,41)(34,35,40,46)(37,47,38,48)(42,44,43,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,37,39,38)(34,42,40,43)(35,44,46,45)(36,47,41,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,37)(34,43,40,42)(35,45,46,44)(36,48,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,d^2=g,e^2=gi,f^2=h,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, 8002615);