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 | |
3884 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)(3,5,6,8)(4,12,15,10)(11,16,13,14)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,39,42)(35,40,45,48)(36,37,38,41)(43,44,46,47);(1,3,7,6)(2,8,9,5)(4,13,15,11)(10,14,12,16)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,39,45)(34,40,42,48)(36,46,38,43)(37,47,41,44);(2,9)(5,8)(10,12)(14,16)(19,24)(21,29)(27,30)(28,32)(33,36,39,38)(34,41,42,37)(35,43,45,46)(40,47,48,44);(1,4,7,15)(2,10,9,12)(3,11,6,13)(5,14,8,16)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,35)(34,40)(36,43)(37,44)(38,46)(39,45)(41,47)(42,48);(1,5,7,8)(2,3,9,6)(4,14,15,16)(10,11,12,13)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,37,39,41)(34,36,42,38)(35,44,45,47)(40,43,48,46);(1,6,7,3)(2,8,9,5)(4,11,15,13)(10,14,12,16)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,38,39,36)(34,41,42,37)(35,43,45,46)(40,44,48,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529284);