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×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,8,10,5)(4,9,13,14)(11,15,16,12)(17,18,23,28)(19,29,30,24)(20,26,21,25)(22,27,32,31)(33,34)(35,36)(37,45)(38,40)(39,42)(41,48)(43,47)(44,46);(1,3)(2,8)(4,11)(5,6)(7,10)(9,15)(12,14)(13,16)(17,19,23,30)(18,24,28,29)(20,27,21,31)(22,25,32,26)(33,35,39,40)(34,36,42,38)(37,46,48,43)(41,47,45,44);(3,10)(5,8)(11,16)(12,15)(17,20,23,21)(18,25,28,26)(19,27,30,31)(22,24,32,29)(33,36,39,38)(34,40,42,35)(37,47,48,44)(41,43,45,46);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,20)(18,26,28,25)(19,27,30,31)(22,24,32,29)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,19)(18,24)(20,31)(21,27)(22,25)(23,30)(26,32)(28,29)(33,37,39,48)(34,41,42,45)(35,43,40,46)(36,44,38,47);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)(17,22,23,32)(18,27,28,31)(19,25,30,26)(20,24,21,29)(33,38,39,36)(34,40,42,35)(37,47,48,44)(41,46,45,43)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541916);