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 | |
3775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(18,20)(19,25)(21,29)(30,32)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(3,9)(6,12)(10,15)(13,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(34,40)(35,42)(41,44)(43,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,20,22,18)(19,23,25,26)(21,28,29,24)(27,30,31,32)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065952);