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 | |
4196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,21)(20,28)(22,24)(23,27)(25,29)(26,31)(30,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,27)(18,23,24,21)(20,26,29,32)(25,31,28,30)(35,41)(38,45)(42,47)(46,48);(2,6)(7,9)(8,12)(14,15)(18,24)(20,29)(21,23)(26,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,25)(19,26)(21,30)(22,29)(23,31)(24,28)(27,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,22,23)(18,19,24,27)(20,30,29,31)(25,26,28,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,30)(29,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150416);