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 | |
3989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,13,16)(11,15,14,12)(17,18)(19,27)(20,23)(21,25)(22,26)(24,30)(28,32)(29,31)(33,34)(35,39)(36,40)(37,48)(38,42)(41,45)(43,44)(46,47);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,35,38,44)(34,39,42,43)(36,45,47,37)(40,41,46,48);(3,5)(6,8)(11,14)(12,15)(21,29)(22,28)(25,31)(26,32)(33,34,38,42)(35,43,44,39)(36,46,47,40)(37,48,45,41);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,20,23,18)(19,27,30,24)(21,25,29,31)(22,32,28,26)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,44)(45,48);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,21,23,29)(18,25,20,31)(19,28,30,22)(24,26,27,32)(33,36)(34,40)(35,37)(38,47)(39,48)(41,43)(42,46)(44,45);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,22,23,28)(18,26,20,32)(19,29,30,21)(24,25,27,31)(33,37)(34,41)(35,36)(38,45)(39,46)(40,43)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=i,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,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9452750);