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 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(4,11)(8,10)(12,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34)(35,36)(37,46)(38,45)(39,44)(40,41)(42,48)(43,47);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(19,23)(20,27)(26,31)(28,32)(33,35)(34,40)(36,44)(37,45)(38,42)(39,41)(43,48)(46,47);(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,36)(34,41)(35,44)(37,47)(38,48)(39,40)(42,43)(45,46);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,37,39,48)(34,42,44,46)(35,45,41,43)(36,47,40,38);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,38,39,47)(34,43,44,45)(35,46,41,42)(36,48,40,37);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,35)(34,40)(36,44)(37,43)(38,46)(39,41)(42,47)(45,48)Polycyclic presentation:
2^9|b^2=gh,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10330191);