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 | |
4054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,5,7)(4,12,14,9)(11,15,13,16)(17,18,21,24)(19,22,26,30)(20,27,28,23)(25,31,32,29)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,3,6,5)(2,7,8,10)(4,13,14,11)(9,16,12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(2,8)(7,10)(9,12)(15,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,4,6,14)(2,9,8,12)(3,11,5,13)(7,15,10,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,14)(13,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,5,6,3)(2,10,8,7)(4,13,14,11)(9,16,12,15)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8026844);