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 | |
3949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,14,16)(4,9,12,13)(5,15,11,6)(17,18)(19,21)(20,24)(22,25)(23,26)(27,29)(28,30)(31,32)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,3)(2,8)(4,5)(6,13)(7,14)(9,15)(10,16)(11,12)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(1,4)(2,9)(3,11)(5,14)(6,8)(7,12)(10,13)(15,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,30,32,27)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,4)(2,9)(3,5)(6,16)(7,12)(8,15)(10,13)(11,14)(17,19)(18,20)(21,26)(22,30)(23,24)(25,29)(27,31)(28,32)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,5)(2,6)(3,12)(4,14)(7,11)(8,9)(10,15)(13,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,30,32,27)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,6)(2,5)(3,13)(4,8)(7,15)(9,14)(10,11)(12,16)(17,22,23,31)(18,25,26,32)(19,27,24,30)(20,28,21,29)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319525);