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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3875 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)(3,11,12,6)(4,13,16,9)(5,15,14,10)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,7,12)(2,6,8,11)(4,14,16,5)(9,10,13,15)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(34,39)(35,40)(41,46)(42,47);(2,8)(6,11)(9,13)(10,15)(19,25)(22,31)(27,29)(28,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4,7,16)(2,9,8,13)(3,5,12,14)(6,15,11,10)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,5)(2,10)(3,4)(6,13)(7,14)(8,15)(9,11)(12,16)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48);(1,6,7,11)(2,3,8,12)(4,15,16,10)(5,13,14,9)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231308);