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 | #2688 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18,23,25)(19,26,22,24)(20,29,30,21)(27,32,31,28)(33,34,39,43)(35,40,44,47)(36,41,46,45)(37,38,42,48);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,23,22)(18,24,25,26)(20,27,30,31)(21,32,29,28)(33,35)(34,40)(36,38)(37,45)(39,44)(41,42)(43,47)(46,48);(2,9)(3,11)(4,13)(10,14)(18,25)(20,30)(24,26)(27,31)(35,44)(36,46)(40,47)(41,45);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,20,23,30)(18,21,25,29)(19,27,22,31)(24,32,26,28)(33,36)(34,41)(35,38)(37,47)(39,46)(40,42)(43,45)(44,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,21)(18,20)(19,28)(22,32)(23,29)(24,31)(25,30)(26,27)(33,37)(34,38)(35,45)(36,40)(39,42)(41,44)(43,48)(46,47);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,22,23,19)(18,26,25,24)(20,31,30,27)(21,28,29,32)(33,38)(34,42)(35,36)(37,43)(39,48)(40,41)(44,46)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034731);