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:
(4,11)(8,13)(9,15)(14,16)(17,18,22,23)(19,20,27,25)(21,29,30,24)(26,31,32,28)(34,38)(35,41)(40,45)(42,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18,22,23)(19,25,27,20)(21,24,30,29)(26,31,32,28)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(2,6)(7,12)(8,13)(14,16)(18,23)(20,25)(24,29)(28,31)(36,43)(40,45)(42,47)(46,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025835);