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 | |
4128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,4,8,7)(5,9,14,16)(11,12,15,13)(17,18,22,26)(19,27,29,23)(20,30,31,24)(21,25,28,32)(33,34)(35,40)(36,38)(37,41)(39,43)(42,48)(44,46)(45,47);(1,3,6,8)(2,4,10,7)(5,11,14,15)(9,12,16,13)(17,19,22,29)(18,23,26,27)(20,21,31,28)(24,25,30,32)(34,36)(39,40)(41,44)(47,48);(1,3)(2,7)(4,10)(5,11)(6,8)(9,13)(12,16)(14,15)(17,20,22,31)(18,24,26,30)(19,28,29,21)(23,32,27,25)(33,35)(34,39)(36,40)(37,42)(38,43)(41,47)(44,48)(45,46);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,20)(18,24)(19,21)(22,31)(23,25)(26,30)(27,32)(28,29)(33,36,38,34)(35,40,43,39)(37,44,46,41)(42,48,45,47);(1,4)(2,8)(3,10)(5,13)(6,7)(9,11)(12,14)(15,16)(17,21)(18,25)(19,20)(22,28)(23,24)(26,32)(27,30)(29,31)(33,35)(34,40)(36,39)(37,45)(38,43)(41,47)(42,46)(44,48);(1,5,6,14)(2,9,10,16)(3,11,8,15)(4,12,7,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,32)(30,31)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10227593);