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 | #53737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,9,11,4)(6,13,14,8)(10,12,16,15)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,39,45)(35,40,44,38)(36,43,47,41)(37,48,46,42);(1,3,7,11)(2,4,5,9)(6,10,14,16)(8,12,13,15)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35,39,44)(34,40,45,38)(36,37,47,46)(41,42,43,48);(2,5)(4,9)(8,13)(12,15)(18,25)(19,30)(20,27)(28,32)(33,36,39,47)(34,41,45,43)(35,46,44,37)(38,42,40,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,37,39,46)(34,42,45,48)(35,36,44,47)(38,43,40,41);(1,5,7,2)(3,4,11,9)(6,8,14,13)(10,15,16,12)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,36)(34,43)(35,46)(37,44)(38,48)(39,47)(40,42)(41,45);(1,6,7,14)(2,8,5,13)(3,10,11,16)(4,12,9,15)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(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=gi,c^2=gh,d^2=i,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=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, 9949789);