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 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,10)(4,7,12,14)(11,13,15,16)(17,18,23,21)(19,25,20,24)(22,29,30,26)(27,31,28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,48)(45,46);(3,9)(4,12)(6,10)(7,14)(18,21)(22,30)(24,25)(27,28)(34,38)(35,37)(41,44)(43,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,45)(34,41,38,44)(35,43,37,46)(40,47,42,48);(1,3)(2,6)(4,11)(5,9)(7,13)(8,10)(12,15)(14,16)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,37,39,35)(34,42,38,40)(36,43,45,46)(41,47,44,48);(1,4,5,12)(2,7,8,14)(3,11,9,15)(6,13,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,34)(35,42,37,40)(36,44,45,41)(43,48,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8849555);