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 | |
4220 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)(19,28,29,24)(20,30,27,22)(21,26,31,32)(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)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(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)(19,29)(20,27)(26,32)(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)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(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)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(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)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=ghi,e^2=gh,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, 10050692);