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 | #2856 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3967 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18)(19,28)(20,21)(22,24)(23,25)(26,27)(29,32)(30,31)(33,34,38,42)(35,36,40,39)(37,45,46,41)(43,47,48,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,22)(18,24,25,28)(20,31,27,29)(21,32,26,30)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(2,6)(7,9)(8,13)(14,15)(18,25)(20,27)(24,28)(29,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,27)(18,26,25,21)(19,29,22,31)(24,32,28,30)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,26)(18,27,25,20)(19,30,22,32)(24,31,28,29)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,47)(46,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,22,23,19)(18,28,25,24)(20,31,27,29)(21,32,26,30)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=h,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, 8022431);