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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,4,11)(6,9,14,13)(10,15,12,16)(17,18)(19,23)(20,30)(21,22)(24,25)(26,31)(27,28)(29,32)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(2,5)(8,11)(9,13)(15,16)(17,19,24,29)(18,23,25,32)(20,27,21,26)(22,31,30,28)(35,41)(39,43)(42,47)(46,48);(1,3,7,4)(2,8,5,11)(6,10,14,12)(9,15,13,16)(17,20,24,21)(18,22,25,30)(19,26,29,27)(23,28,32,31)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,4,7,3)(2,8,5,11)(6,10,14,12)(9,16,13,15)(17,21,24,20)(18,22,25,30)(19,27,29,26)(23,28,32,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,7,2)(3,8,4,11)(6,13,14,9)(10,15,12,16)(17,22)(18,20)(19,28)(21,25)(23,26)(24,30)(27,32)(29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,6,7,14)(2,9,5,13)(3,10,4,12)(8,15,11,16)(17,23,24,32)(18,19,25,29)(20,28,21,31)(22,26,30,27)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8547098);