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 | |
3770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,24,28)(19,23,21,26)(20,30,32,25)(22,27,29,31)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,14)(12,16)(17,19,24,21)(18,23,28,26)(20,22,32,29)(25,31,30,27)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(2,6)(3,9)(4,11)(15,16)(17,20,24,32)(18,25,28,30)(19,29,21,22)(23,27,26,31)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,3,5,9)(2,7,6,13)(4,12,11,10)(8,16,14,15)(17,21,24,19)(18,26,28,23)(20,29,32,22)(25,27,30,31)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4,5,11)(2,8,6,14)(3,10,9,12)(7,15,13,16)(17,22,24,29)(18,27,28,31)(19,20,21,32)(23,30,26,25)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,14)(12,16)(17,23,24,26)(18,19,28,21)(20,31,32,27)(22,30,29,25)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8916416);