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 |
| prime radical | #58800 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,4,11,7)(5,13,15,9)(10,16,14,12)(18,23)(19,25)(26,31)(27,32)(34,39)(35,42)(37,45)(44,47);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,8)(4,7)(9,13)(12,16)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619175);