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 | |
3976 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18,23,26)(19,24,30,32)(20,25,28,29)(21,31,27,22)(33,34,39,42)(35,43,46,40)(36,38,44,48)(37,41,47,45);(2,8)(5,11)(9,15)(13,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,31)(26,32)(27,28);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,20)(18,25)(19,27)(21,30)(22,24)(23,28)(26,29)(31,32)(33,35,39,46)(34,40,42,43)(36,37,44,47)(38,45,48,41);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,20)(18,25)(19,21)(22,32)(23,28)(24,31)(26,29)(27,30)(33,36)(34,38)(35,37)(39,44)(40,45)(41,43)(42,48)(46,47);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,21)(18,22)(19,28)(20,30)(23,27)(24,25)(26,31)(29,32)(33,37)(34,41)(35,44)(36,46)(38,40)(39,47)(42,45)(43,48);(1,6,7,14)(2,9,8,15)(3,10,4,12)(5,13,11,16)(17,22)(18,21)(19,29)(20,24)(23,31)(25,30)(26,27)(28,32)(33,38)(34,36)(35,45)(37,40)(39,48)(41,46)(42,44)(43,47)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,d^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, 10319396);