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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,13,16)(11,15,14,12)(17,18)(19,26)(20,25)(21,30)(22,27)(23,28)(24,31)(29,32)(33,34)(35,41)(36,46)(37,43)(38,44)(39,40)(42,47)(45,48);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,19)(18,24)(20,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,35,40,37)(34,41,39,43)(36,38,47,45)(42,48,46,44);(3,5)(6,8)(11,14)(12,15)(17,20,23,21)(18,25,28,30)(19,27,31,29)(22,24,32,26)(33,36,40,47)(34,42,39,46)(35,45,37,38)(41,44,43,48);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,21,23,20)(18,25,28,30)(19,29,31,27)(22,24,32,26)(33,37,40,35)(34,43,39,41)(36,45,47,38)(42,44,46,48);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,19)(18,26)(20,29)(21,27)(22,30)(23,31)(24,28)(25,32)(33,38,40,45)(34,44,39,48)(35,36,37,47)(41,46,43,42);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,22,23,32)(18,27,28,29)(19,30,31,25)(20,26,21,24)(33,39,40,34)(35,43,37,41)(36,42,47,46)(38,48,45,44)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gh,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=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, 9453250);