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 | |
4126 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)(3,8,11,15)(4,9,13,12)(5,10,16,14)(19,25)(23,30)(26,28)(31,32)(33,34)(35,37)(36,41)(38,40)(39,42)(43,45)(44,46)(47,48);(2,7)(4,13)(8,15)(10,14)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,38,42)(34,39,40,37)(36,43,46,48)(41,47,44,45);(3,11)(5,16)(8,15)(10,14)(18,22)(19,25)(24,27)(26,28)(34,40)(37,39)(41,44)(45,47);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,34)(35,39)(36,44)(37,42)(38,40)(41,46)(43,45)(47,48);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,47)(40,44)(42,48);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37,38,39)(34,42,40,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838303);