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) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18)(19,20)(21,29)(22,24)(23,31)(25,27)(26,30)(28,32)(33,34)(35,41)(36,45)(37,39)(38,42)(40,43)(44,46)(47,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,30)(31,32)(33,35,40,37)(34,41,43,39)(36,44,38,47)(42,48,45,46);(2,6)(7,9)(8,12)(14,15)(17,18,22,24)(19,25,27,20)(21,23,31,29)(26,30,32,28)(33,36,40,38)(34,42,43,45)(35,44,37,47)(39,46,41,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,25)(20,27)(21,23)(22,24)(26,30)(28,32)(29,31)(33,37,40,35)(34,39,43,41)(36,44,38,47)(42,48,45,46);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20,22,25)(18,19,24,27)(21,30,31,28)(23,32,29,26)(33,38,40,36)(34,42,43,45)(35,44,37,47)(39,48,41,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21)(18,23)(19,26)(20,28)(22,31)(24,29)(25,30)(27,32)(33,39)(34,35)(36,46)(37,43)(38,48)(40,41)(42,44)(45,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=hi,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, 10154483);