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 | |
4199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,23,25)(19,24,28,32)(20,21,26,31)(22,30,29,27)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(4,11)(7,9)(12,16)(14,15)(17,19,23,28)(18,24,25,32)(20,29,26,22)(21,27,31,30)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,3)(7,9)(8,10)(14,15)(18,25)(19,28)(20,26)(27,30)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20)(18,21)(19,29)(22,28)(23,26)(24,27)(25,31)(30,32)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,26)(19,30)(20,25)(22,32)(23,31)(24,29)(27,28)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,29)(18,27,25,30)(19,20,28,26)(21,24,31,32)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=ghi,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, 10080722);