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 | #47535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(33,34)(35,43)(36,46)(37,48)(38,39)(40,45)(41,47)(42,44);(1,2,6,3)(4,9,12,7)(5,10,13,8)(11,14,16,15)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,35)(34,40)(36,44)(37,47)(38,43)(39,45)(41,42)(46,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34,39,38)(35,40,45,43)(36,46,47,41)(37,42,44,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,36,39,47)(34,41,38,46)(35,44,45,37)(40,42,43,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,37,39,44)(34,42,38,48)(35,36,45,47)(40,46,43,41);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,38,39,34)(35,40,45,43)(36,46,47,41)(37,48,44,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8780973);