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 | |
4146 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32);(2,7)(4,15)(8,11)(10,13)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(2,7)(3,12)(4,15)(5,16)(20,31)(21,30)(22,28)(25,32)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,d^2=gi,e^2=h,f^2=g,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, 8901527);