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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3752 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)(3,11)(4,10)(5,9)(6,12)(7,8)(13,16)(14,15)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(35,41)(36,43)(38,44)(39,46);(2,8)(4,14)(9,11)(12,13)(18,22)(19,26)(24,28)(27,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(1,3,7,5)(2,9,8,11)(4,12,14,13)(6,10,16,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,7,14)(2,10,8,15)(3,13,5,12)(6,11,16,9)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,5,7,3)(2,11,8,9)(4,12,14,13)(6,10,16,15)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(1,6,7,16)(2,12,8,13)(3,10,5,15)(4,9,14,11)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=h,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, 8657978);