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 | #2543 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3906 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)(3,10)(4,9)(5,14)(6,7)(8,13)(11,12)(15,16)(20,28)(23,29)(27,31)(30,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,15)(14,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,30)(31,32)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(1,4,7,14)(2,9,6,5)(3,11,13,16)(8,15,10,12)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,30,29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,5)(2,4)(3,12)(6,14)(7,9)(8,11)(10,16)(13,15)(17,18)(19,21)(20,23)(22,25)(24,26)(27,30)(28,29)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,6,7,2)(3,8,13,10)(4,5,14,9)(11,15,16,12)(17,20)(18,23)(19,27)(21,30)(22,28)(24,31)(25,29)(26,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,3)(2,10)(4,11)(5,15)(6,8)(7,13)(9,12)(14,16)(17,21)(18,24)(19,25)(20,30)(22,26)(23,31)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8386743);