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 | |
3965 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,5,6,14)(2,9,10,16)(3,11,8,13)(4,12,7,15)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(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=gh,c^2=h,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, 10319523);