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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,14,8)(11,16,13,15)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,42)(36,41)(37,45)(38,43)(39,44)(40,46)(47,48);(2,6)(3,9)(8,12)(11,13)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35)(34,40)(36,43)(37,47)(38,45)(39,46)(41,48)(42,44);(2,6)(7,10)(8,12)(15,16)(18,25)(19,20)(26,32)(27,28)(33,36,39,37)(34,41,44,45)(35,43,46,47)(38,40,48,42);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,14)(13,15)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,37,39,36)(34,41,44,45)(35,43,46,47)(38,42,48,40);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,35)(34,42)(36,43)(37,47)(38,41)(39,46)(40,44)(45,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,13)(10,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,38)(34,43)(35,45)(36,40)(37,42)(39,48)(41,46)(44,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154449);