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 | |
3871 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)(3,8)(4,5)(6,14)(7,9)(10,16)(11,13)(12,15)(17,18)(19,24)(20,22)(21,28)(23,27)(25,30)(26,31)(29,32)(34,39)(35,42)(41,48)(43,44);(1,3,7,4)(2,8,9,5)(6,11,16,12)(10,15,14,13)(17,19,22,27)(18,23,20,24)(21,26,30,29)(25,31,28,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,45)(44,47)(46,48);(2,9)(3,4)(10,14)(11,12)(18,20)(23,24)(25,28)(31,32);(1,4,7,3)(2,5,9,8)(6,12,16,11)(10,13,14,15)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,47,48,45);(1,5)(2,4)(3,9)(6,15)(7,8)(10,11)(12,14)(13,16)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32)(33,36,38,46)(34,41,39,48)(35,43,42,44)(37,47,40,45);(1,6,7,16)(2,10,9,14)(3,11,4,12)(5,13,8,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,43,48,44)Polycyclic presentation:
2^9|b^2=h,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10075552);