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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,25)(19,23,27,20)(21,24,30,32)(26,29,31,28)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(1,2)(3,8)(4,5)(6,7)(9,14)(10,16)(11,13)(12,15)(17,19,22,27)(18,20,25,23)(21,26,30,31)(24,28,32,29)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(2,7)(5,9)(8,15)(11,16)(17,20,22,23)(18,19,25,27)(21,29,30,28)(24,31,32,26)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,4)(2,9)(3,10)(5,7)(6,14)(8,11)(12,13)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,29)(28,32)(30,31)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44);(1,5,6,9)(2,4,7,14)(3,11,12,16)(8,13,15,10)(17,21,22,30)(18,24,25,32)(19,26,27,31)(20,28,23,29)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8279334);