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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,10,4)(5,13,15,9)(11,12,16,14)(17,18,23,25)(19,24,28,21)(20,31,30,26)(22,32,29,27)(33,34,39,36)(35,43,44,40)(37,41,48,45)(38,42,47,46);(2,7)(3,10)(9,13)(11,16)(17,19,23,28)(18,24,25,21)(20,22,30,29)(26,27,31,32)(33,35,39,44)(34,40,36,43)(37,47,48,38)(41,42,45,46);(2,7)(3,10)(9,13)(11,16)(18,25)(19,28)(20,30)(27,32);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,20,23,30)(18,26,25,31)(19,29,28,22)(21,27,24,32)(33,36,39,34)(35,40,44,43)(37,45,48,41)(38,46,47,42);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21)(18,19)(20,27)(22,31)(23,24)(25,28)(26,29)(30,32)(33,37,39,48)(34,41,36,45)(35,38,44,47)(40,46,43,42);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,22,23,29)(18,27,25,32)(19,30,28,20)(21,26,24,31)(33,38)(34,42)(35,37)(36,46)(39,47)(40,45)(41,43)(44,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=gh,e^2=g,f^2=h,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, 9845346);