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 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,24)(20,29)(21,26)(22,31)(23,25)(27,28)(30,32)(34,36)(35,38)(41,44)(43,45);(4,11)(8,14)(10,15)(12,16)(17,19)(18,24)(20,30)(21,23)(22,28)(25,26)(27,31)(29,32)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,7)(3,9)(8,14)(10,15)(18,25)(19,21)(20,31)(28,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20)(18,22)(19,27)(21,30)(23,31)(24,32)(25,29)(26,28)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,19)(18,26,25,24)(20,30,31,27)(22,28,29,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22)(18,20)(19,28)(21,32)(23,29)(24,30)(25,31)(26,27)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962627);