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 | #45066 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3913 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)(4,12)(5,9)(7,11)(8,13)(10,15)(14,16)(18,20)(19,21)(22,30)(31,32)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,18)(19,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(1,2,5,9)(3,7,11,6)(4,8,13,12)(10,15,16,14)(17,19,23,21)(18,24,20,27)(22,26,30,29)(25,31,28,32)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,2)(3,7)(4,8)(5,9)(6,11)(10,15)(12,13)(14,16)(17,20,23,18)(19,24,21,27)(22,28,30,25)(26,31,29,32)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21,23,19)(18,24,20,27)(22,29,30,26)(25,31,28,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22,23,30)(18,25,20,28)(19,26,21,29)(24,31,27,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022585);