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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,11,13,7)(4,14,15,8)(5,9,12,16)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3,6,13)(2,7,10,11)(4,12,15,5)(8,16,14,9)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(34,39)(35,42)(41,44)(43,48);(4,15)(5,12)(8,14)(9,16)(18,21)(20,30)(24,26)(27,29)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,4)(2,8)(3,12)(5,13)(6,15)(7,16)(9,11)(10,14)(17,20)(18,25)(19,27)(21,28)(22,29)(23,30)(24,31)(26,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,5,6,12)(2,9,10,16)(3,4,13,15)(7,8,11,14)(17,21,23,18)(19,26,22,24)(20,28,30,25)(27,32,29,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,2)(3,7)(4,8)(5,9)(6,10)(11,13)(12,16)(14,15)(17,22,23,19)(18,26,21,24)(20,29,30,27)(25,32,28,31)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034773);