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 | |
4133 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(4,11)(8,14)(10,15)(12,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,27)(28,30)(31,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,7)(3,9)(8,14)(10,15)(17,19,22,29)(18,23,27,24)(20,28,30,26)(21,25,31,32)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19)(18,24)(20,28)(21,25)(22,29)(23,27)(26,30)(31,32)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,25)(19,28)(21,24)(22,30)(23,31)(26,29)(27,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,26)(19,25)(20,23)(22,31)(24,30)(27,28)(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=i,d^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9877339);