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 | |
4185 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,26)(21,22)(23,27)(25,31)(28,30)(29,32)(33,34)(35,42)(36,45)(37,40)(38,44)(39,41)(43,47)(46,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,23,22)(18,24,27,21)(20,29,31,28)(25,30,26,32)(33,35)(34,38)(36,46)(37,43)(39,44)(40,48)(41,42)(45,47);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,32)(26,27)(33,36,39,37)(34,40,41,45)(35,43,44,46)(38,47,42,48);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,20)(18,26)(19,29)(21,30)(22,28)(23,31)(24,32)(25,27)(33,35)(34,38)(36,43)(37,46)(39,44)(40,47)(41,42)(45,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,21,23,24)(18,19,27,22)(20,30,31,32)(25,28,26,29)(33,37,39,36)(34,40,41,45)(35,43,44,46)(38,48,42,47);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,22,23,19)(18,24,27,21)(20,29,31,28)(25,32,26,30)(33,38)(34,35)(36,47)(37,48)(39,42)(40,43)(41,44)(45,46)Polycyclic presentation:
2^9|b^2=gh,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10353683);