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 | |
4262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34)(35,41)(36,37)(38,42)(39,45)(40,46)(43,47)(44,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(2,9)(3,4)(10,16)(11,12)(18,24)(20,25)(23,30)(26,31);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,36,38,46)(34,40,42,37)(35,43,45,48)(39,47,41,44);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,38,40)(34,36,42,46)(35,44,45,47)(39,43,41,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,35)(34,41)(36,43)(37,47)(38,45)(39,42)(40,44)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=ghi,e^2=g,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, 10028443);