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 | |
4171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,27)(20,24)(21,30)(22,26)(23,29)(25,31)(28,32)(33,34)(35,38)(36,39)(37,45)(40,42)(41,48)(43,44)(46,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,29)(18,23,26,27)(20,21,28,31)(24,25,32,30)(33,35,40,39)(34,38,42,36)(37,43,48,47)(41,46,45,44);(2,3)(7,9)(8,10)(14,15)(17,18,22,26)(19,23,29,27)(20,24,28,32)(21,25,31,30)(33,36,40,38)(34,35,42,39)(37,46,48,44)(41,47,45,43);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,32)(30,31)(33,37,40,48)(34,41,42,45)(35,43,39,47)(36,44,38,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,24)(19,21)(22,28)(23,25)(26,32)(27,30)(29,31)(33,38,40,36)(34,35,42,39)(37,46,48,44)(41,43,45,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,25)(19,28)(20,29)(22,31)(23,32)(24,27)(26,30)(33,39,40,35)(34,38,42,36)(37,47,48,43)(41,46,45,44)Polycyclic presentation:
2^9|b^2=h,c^2=gi,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9831104);