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 | |
4066 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(21,29)(26,30)(27,32)(28,31)(33,34,40,36)(35,37,42,39)(38,41,47,44)(43,45,48,46);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35)(34,39)(36,37)(38,43)(40,42)(41,46)(44,45)(47,48);(2,7)(6,13)(8,11)(14,15)(18,23)(20,24)(26,30)(28,31)(33,36,40,34)(35,37,42,39)(38,41,47,44)(43,46,48,45);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19,22,25)(18,24,23,20)(21,27,29,32)(26,31,30,28)(33,37,40,39)(34,35,36,42)(38,45,47,46)(41,43,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,38,40,47)(34,41,36,44)(35,43,42,48)(37,45,39,46);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,39,40,37)(34,42,36,35)(38,46,47,45)(41,48,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002658);