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 | #2540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3871 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)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(3,4)(7,10)(11,12)(15,16)(17,18)(19,23)(20,22)(21,25)(24,27)(26,30)(28,29)(31,32)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,20,22,18)(19,23,27,24)(21,25,29,28)(26,31,32,30)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,40)(34,44,45,35)(36,42,47,48)(37,43,46,41)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9954524);