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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,15)(4,11,12,8)(10,16,13,14)(17,18)(19,25)(20,23)(21,29)(22,24)(26,30)(27,31)(28,32)(33,34,38,40)(35,39,43,48)(36,44,46,41)(37,47,45,42);(2,7)(3,9)(4,12)(14,16)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,35,38,43)(34,39,40,48)(36,45,46,37)(41,47,44,42);(3,9)(5,15)(10,13)(14,16)(18,24)(21,30)(23,25)(27,28)(34,40)(35,43)(36,46)(42,47);(1,3,6,9)(2,5,7,15)(4,13,12,10)(8,16,11,14)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,36,38,46)(34,41,40,44)(35,37,43,45)(39,42,48,47);(1,4,6,12)(2,8,7,11)(3,10,9,13)(5,14,15,16)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,37)(34,42)(35,36)(38,45)(39,41)(40,47)(43,46)(44,48);(1,5)(2,9)(3,7)(4,14)(6,15)(8,13)(10,11)(12,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,48)(44,46)(45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840348);