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 | #49489 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4100 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34)(35,38)(36,41)(37,42)(39,40)(43,48)(44,46)(45,47);(2,7)(4,8)(9,13)(12,14)(20,27)(24,30)(26,31)(28,32)(33,35,39,43)(34,38,40,48)(36,44,45,37)(41,46,47,42);(2,7)(3,10)(9,13)(11,16)(18,23)(19,25)(24,30)(26,31)(34,40)(35,43)(36,45)(42,46);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36)(34,41)(35,44)(37,43)(38,46)(39,45)(40,47)(42,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,39,44)(34,42,40,46)(35,45,43,36)(38,47,48,41);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38)(34,43)(35,40)(36,46)(37,41)(39,48)(42,45)(44,47)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10078866);