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 | |
4210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(5,13)(7,9)(11,16)(20,28)(24,31)(26,32)(29,30)(33,34)(35,42)(36,39)(37,44)(38,40)(41,46)(43,47)(45,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34,39,36)(35,42,38,40)(37,41,46,44)(43,48,45,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044423);