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 | |
4301 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)(3,7)(4,8)(5,11)(6,9)(10,13)(12,15)(14,16)(17,18,24,27)(19,28,22,25)(20,23,21,26)(29,30,31,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,48);(1,3,6,11)(2,7,9,5)(4,10,14,12)(8,13,16,15)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32)(33,35,39,40)(34,37,42,36)(38,43,47,48)(41,45,46,44);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,48,46,43);(1,3)(2,5)(4,12)(6,11)(7,9)(8,13)(10,14)(15,16)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,48,46,43);(1,4,6,14)(2,8,9,16)(3,10,11,12)(5,15,7,13)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,44)(45,48);(1,5)(2,3)(4,13)(6,7)(8,12)(9,11)(10,16)(14,15)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411804);