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 | #35696 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4089 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)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(18,23)(19,24)(25,30)(26,31)(33,34)(35,37)(36,45)(38,46)(39,41)(40,44)(42,43)(47,48);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35,39,44)(34,40,41,37)(36,42,46,48)(38,43,45,47);(2,9)(3,4)(5,15)(6,16)(19,24)(20,27)(21,29)(25,30)(33,36,39,46)(34,38,41,45)(35,42,44,48)(37,47,40,43);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36)(34,38)(35,42)(37,47)(39,46)(40,43)(41,45)(44,48);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,39,40)(34,35,41,44)(36,43,46,47)(38,48,45,42);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38)(34,36)(35,43)(37,48)(39,45)(40,42)(41,46)(44,47)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=i,e^2=h,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, 8901550);