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 | |
4181 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)(3,8)(4,14)(5,9)(6,10)(7,13)(11,12)(15,16)(17,18)(19,25)(20,28)(21,27)(22,23)(24,29)(26,31)(30,32)(33,34,39,41)(35,40,45,47)(36,42,46,48)(37,43,44,38);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(17,19,22,27)(18,21,23,25)(20,26,29,30)(24,31,28,32)(33,35)(34,40)(36,44)(37,46)(38,42)(39,45)(41,47)(43,48);(4,9)(5,14)(11,15)(12,16)(18,23)(20,29)(21,25)(26,30)(34,41)(35,45)(36,46)(38,43);(1,4,6,9)(2,5,10,14)(3,11,13,15)(7,12,8,16)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,32)(27,30)(33,36,39,46)(34,42,41,48)(35,44,45,37)(38,47,43,40);(1,3)(2,8)(4,15)(5,12)(6,13)(7,10)(9,11)(14,16)(17,19)(18,25)(20,30)(21,23)(22,27)(24,31)(26,29)(28,32)(33,37,39,44)(34,43,41,38)(35,36,45,46)(40,42,47,48);(1,5)(2,9)(3,12)(4,10)(6,14)(7,15)(8,11)(13,16)(17,21,22,25)(18,19,23,27)(20,31,29,32)(24,26,28,30)(33,38,39,43)(34,44,41,37)(35,42,45,48)(36,47,46,40)Polycyclic presentation:
2^9|a^2=i,b^2=hi,d^2=gh,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8827927);