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 | |
4153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(5,13)(12,15)(17,18,24,27)(19,25,29,31)(20,30,21,26)(22,23,32,28)(33,34,39,42)(35,36,40,37)(38,41,46,48)(43,44,47,45);(1,2,6,7)(3,4,9,8)(5,10,13,14)(11,12,16,15)(17,19,24,29)(18,25,27,31)(20,28,21,23)(22,26,32,30)(33,35,39,40)(34,37,42,36)(38,43,46,47)(41,45,48,44);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20,24,21)(18,26,27,30)(19,28,29,23)(22,25,32,31)(33,36,39,37)(34,40,42,35)(38,44,46,45)(41,47,48,43);(1,3)(2,4)(5,11)(6,9)(7,8)(10,12)(13,16)(14,15)(17,21,24,20)(18,26,27,30)(19,23,29,28)(22,25,32,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4)(2,9)(3,7)(5,12)(6,8)(10,16)(11,14)(13,15)(17,22)(18,23)(19,26)(20,25)(21,31)(24,32)(27,28)(29,30)(33,37,39,36)(34,35,42,40)(38,44,46,45)(41,47,48,43);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,23)(18,22)(19,20)(21,29)(24,28)(25,26)(27,32)(30,31)(33,38,39,46)(34,41,42,48)(35,43,40,47)(36,44,37,45)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,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^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10428554);