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 | #53088 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3995 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)(4,9)(5,10)(6,8)(7,14)(12,16)(13,15)(17,18)(19,24)(20,30)(21,25)(22,26)(23,27)(28,32)(29,31)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,13,16,9)(17,19)(18,24)(20,22)(21,31)(23,28)(25,29)(26,30)(27,32)(35,41)(38,45)(42,47)(46,48);(2,8)(5,16)(7,11)(9,13)(19,28)(22,29)(24,32)(26,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,11,16,7)(17,20)(18,21)(19,29)(22,28)(23,25)(24,26)(27,30)(31,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,3)(2,7)(4,10)(5,13)(6,14)(8,11)(9,16)(12,15)(17,21)(18,25)(19,26)(20,27)(22,24)(23,30)(28,31)(29,32)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,11,15,7)(17,22,23,29)(18,26,27,31)(19,20,28,25)(21,24,30,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8325320);