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 | |
4209 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,30)(21,26)(22,23)(24,28)(27,29)(31,32)(33,34)(35,40)(36,37)(38,41)(39,42)(43,46)(44,45)(47,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,24,31)(18,25,28,32)(20,29,21,23)(22,30,27,26)(33,34,39,42)(35,40,37,36)(38,41,47,48)(43,46,45,44);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20,24,21)(18,26,28,30)(19,29,31,23)(22,32,27,25)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,21,24,20)(18,26,28,30)(19,23,31,29)(22,32,27,25)(33,36)(34,35)(37,42)(38,46)(39,40)(41,45)(43,48)(44,47);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,22,24,27)(18,23,28,29)(19,30,31,26)(20,32,21,25)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,46,48,44);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,23)(18,27)(19,20)(21,31)(22,28)(24,29)(25,26)(30,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=gh,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, 10316979);