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 | |
3972 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)(3,11)(4,8)(5,6)(7,10)(9,16)(12,15)(13,14)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,3,7,4)(2,8,10,11)(5,12,9,14)(6,13,16,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,36,39,37)(34,40,42,35)(38,44,47,45)(41,46,48,43);(1,4,7,3)(2,8,10,11)(5,12,9,14)(6,15,16,13)(17,18)(19,20)(21,29)(22,25)(23,26)(24,30)(27,32)(28,31)(33,35)(34,36)(37,42)(38,46)(39,40)(41,45)(43,47)(44,48);(1,5,7,9)(2,6,10,16)(3,12,4,14)(8,13,11,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,37,39,36)(34,40,42,35)(38,45,47,44)(41,46,48,43);(1,6)(2,9)(3,13)(4,15)(5,10)(7,16)(8,14)(11,12)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38,39,47)(34,41,42,48)(35,43,40,46)(36,44,37,45)Polycyclic presentation:
2^9|c^2=gh,d^2=h,e^2=gh,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, 10316881);