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 | |
4227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,11)(4,10,15,14)(6,13,16,12)(18,23)(19,25)(26,30)(27,32)(33,34,38,40)(35,43,44,39)(36,45,46,41)(37,42,47,48);(1,3)(2,8)(4,13)(5,7)(6,14)(9,11)(10,16)(12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,38,44)(34,39,40,43)(36,37,46,47)(41,42,45,48);(2,9)(3,5)(4,15)(6,16)(19,25)(20,24)(21,29)(26,30)(34,40)(36,46)(37,47)(39,43);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,36,38,46)(34,41,40,45)(35,37,44,47)(39,42,43,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,37)(34,42)(35,36)(38,47)(39,41)(40,48)(43,45)(44,46);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=h,e^2=gh,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, 8886976);