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 | |
3963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,12)(5,8,15,16)(10,11,13,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,30)(27,32)(29,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(2,7)(3,9)(8,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(35,41)(38,45)(42,47)(46,48);(2,7)(3,9)(8,16)(10,13)(18,19)(23,25)(24,26)(31,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,4)(5,13)(6,9)(7,12)(8,14)(10,15)(11,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,4)(2,3)(5,14)(6,12)(7,9)(8,13)(10,16)(11,15)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,5,6,15)(2,8,7,16)(3,10,9,13)(4,11,12,14)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657969);