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 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,9,14,15)(11,12,13,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(2,7)(3,10)(9,15)(11,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(35,36)(39,42)(43,44)(46,47);(2,7)(4,8)(9,15)(12,16)(21,29)(24,28)(26,32)(30,31)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,48,47);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,14)(15,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36,38,35)(34,39,41,42)(37,43,45,44)(40,47,48,46);(1,4,6,8)(2,3,7,10)(5,12,14,16)(9,11,15,13)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,46)(45,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,19)(18,25)(20,23)(21,26)(22,27)(24,31)(28,30)(29,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442461);