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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,22,25)(19,23,26,21)(20,24,28,32)(27,31,29,30)(33,34,38,40)(35,43,44,39)(36,45,46,41)(37,42,47,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,31)(28,29)(30,32)(33,35,38,44)(34,39,40,43)(36,37,46,47)(41,42,45,48);(2,3)(4,11)(5,13)(14,15)(19,26)(20,28)(21,23)(24,32)(34,40)(36,46)(37,47)(39,43);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,29)(21,25)(22,26)(24,30)(27,28)(31,32)(33,36,38,46)(34,41,40,45)(35,37,44,47)(39,42,43,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,37)(34,42)(35,36)(38,47)(39,41)(40,48)(43,45)(44,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,23)(18,19,25,26)(20,30,28,31)(24,27,32,29)(33,34)(35,39)(36,41)(37,42)(38,40)(43,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8927521);