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 | |
3977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,10,11)(5,8,14,12)(9,15,13,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,31)(26,28)(30,32)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(2,4)(7,11)(8,12)(15,16)(17,19,22,27)(18,23,25,20)(21,26,31,32)(24,30,29,28)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(1,3,6,10)(2,7,4,11)(5,13,14,9)(8,16,12,15)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,15)(12,16)(13,14)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,4,6,2)(3,7,10,11)(5,12,14,8)(9,15,13,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,14)(2,8,4,12)(3,9,10,13)(7,15,11,16)(17,21,22,31)(18,24,25,29)(19,26,27,32)(20,28,23,30)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8957589);