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 | #51968 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,11,14,8)(4,9,13,15)(5,10,12,16)(17,18,22,27)(19,26,29,23)(20,30,28,24)(21,25,31,32)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,48)(44,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(17,19,22,29)(18,23,27,26)(20,21,28,31)(24,25,30,32)(34,39)(36,41)(40,42)(43,47);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,20,22,28)(18,24,27,30)(19,21,29,31)(23,25,26,32)(33,34,38,39)(35,40,45,42)(36,37,41,46)(43,44,47,48);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,10)(11,16)(17,21,22,31)(18,25,27,32)(19,28,29,20)(23,30,26,24)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,19)(18,26)(20,21)(22,29)(23,27)(24,32)(25,30)(28,31)(33,36,38,41)(34,37,39,46)(35,43,45,47)(40,44,42,48);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,10,12,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,29)(28,30)(31,32)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9113793);