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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(18,23)(19,24)(21,29)(27,32)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,31)(30,32)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(2,6)(7,13)(8,14)(15,16)(17,19,22,24)(18,20,23,28)(21,30,29,26)(25,32,31,27)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,31)(30,32)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,30)(26,31)(29,32)(33,37,39,46)(34,41,36,44)(35,42,43,48)(38,45,40,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,21,22,29)(18,25,23,31)(19,26,24,30)(20,27,28,32)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8808690);