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 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(2,8)(3,4)(10,15)(11,12)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=ghi,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,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066469);