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 | #51291 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3991 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)(3,8,5,11)(4,10,15,14)(6,13,16,12)(17,18,22,24)(19,25,26,21)(20,23,28,31)(27,32,29,30)(33,34,39,42)(35,40,38,36)(37,41,47,48)(43,45,46,44);(1,3)(2,8)(4,13)(5,7)(6,14)(9,11)(10,16)(12,15)(17,18)(19,21)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,35,39,38)(34,40,42,36)(37,43,47,46)(41,45,48,44);(2,9)(3,5)(4,15)(6,16)(19,26)(20,28)(21,25)(23,31)(33,36,39,40)(34,35,42,38)(37,45,47,44)(41,46,48,43);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19)(18,21)(20,29)(22,26)(23,32)(24,25)(27,28)(30,31)(33,36)(34,35)(37,45)(38,42)(39,40)(41,46)(43,48)(44,47);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,32)(26,29)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,32)(29,31)(33,38,39,35)(34,36,42,40)(37,46,47,43)(41,44,48,45)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8937229);