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 | |
4085 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)(3,8)(4,6)(5,10)(7,9)(11,13)(12,15)(14,16)(18,23)(19,25)(26,31)(27,32)(33,34,37,41)(35,40,43,38)(36,44,45,39)(42,46,48,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,15,16,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,35,37,43)(34,38,41,40)(36,42,45,48)(39,46,44,47);(2,9)(3,4)(10,16)(11,12)(33,36,37,45)(34,39,41,44)(35,42,43,48)(38,46,40,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,15,16,13)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(33,36)(34,39)(35,42)(37,45)(38,46)(40,47)(41,44)(43,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,15,8,13)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,46)(45,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,12)(11,16)(14,15)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619187);