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 | |
3817 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)(3,8)(4,12)(5,10)(6,7)(9,14)(11,13)(15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32)(34,38)(36,43)(39,44)(42,47);(2,7)(3,11)(9,12)(10,16)(17,19)(18,23)(20,25)(21,28)(22,27)(24,32)(26,30)(29,31)(35,41)(39,44)(42,47)(46,48);(2,7)(3,11)(9,12)(10,16)(19,27)(20,29)(23,30)(24,28)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,9,15,12)(17,20,22,29)(18,24,26,28)(19,25,27,31)(21,23,32,30)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4,6,14)(2,9,7,12)(3,10,11,16)(5,8,15,13)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,30)(28,29)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,5,6,15)(2,10,7,16)(3,12,11,9)(4,13,14,8)(17,21,22,32)(18,25,26,31)(19,28,27,24)(20,30,29,23)Polycyclic presentation:
2^9|c^2=gh,d^2=gi,e^2=g,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, 8657583);