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 | |
4247 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)(3,9)(4,8)(5,10)(6,7)(11,12)(13,15)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,2)(3,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(19,25)(22,29)(26,31)(30,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,7)(3,10)(8,15)(11,16)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,3)(2,5)(4,11)(6,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,4,6,13)(2,8,7,15)(3,11,10,16)(5,14,9,12)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,5)(2,3)(4,12)(6,9)(7,10)(8,16)(11,15)(13,14)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9822534);