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 | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(18,21)(19,20)(26,29)(27,28);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,45)(46,48);(4,11)(7,9)(12,16)(14,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35,38,40)(34,37,41,42)(36,44,46,43)(39,47,48,45);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,47)(46,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,36,38,46)(34,39,41,48)(35,43,40,44)(37,45,42,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,37)(34,40)(35,41)(36,45)(38,42)(39,44)(43,48)(46,47)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441935);