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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3955 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18)(19,27)(20,21)(22,25)(23,24)(26,31)(28,29)(30,32)(35,41)(36,43)(38,44)(39,46);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(18,24)(20,26)(25,27)(28,30)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(3,11)(4,13)(7,14)(8,12)(17,19,23,22)(18,25,24,27)(20,30,26,28)(21,29,31,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,20,23,26)(18,21,24,31)(19,28,22,30)(25,32,27,29)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,21)(18,26)(19,29)(20,24)(22,32)(23,31)(25,28)(27,30)(33,36,37,43)(34,39,40,46)(35,42,41,47)(38,45,44,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,22,23,19)(18,25,24,27)(20,28,26,30)(21,29,31,32)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657698);