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 | |
4156 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)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,29,28,31)(33,34,39,41)(35,44,46,40)(36,37,42,45)(38,47,48,43);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,27)(18,21,25,23)(20,28,30,26)(24,29,32,31)(33,35,39,46)(34,40,41,44)(36,47,42,43)(37,38,45,48);(2,3)(7,9)(8,10)(14,15)(34,41)(36,42)(40,44)(43,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,30)(31,32)(33,36,39,42)(34,37,41,45)(35,43,46,47)(38,40,48,44);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,31)(22,30)(23,29)(25,32)(27,28)(33,37)(34,42)(35,38)(36,41)(39,45)(40,43)(44,47)(46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,28,32,26)(33,38)(34,43)(35,45)(36,40)(37,46)(39,48)(41,47)(42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9845264);