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 | |
4005 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)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18)(19,29)(20,26)(21,22)(23,28)(24,31)(25,32)(27,30)(33,34,39,38)(35,42,36,40)(37,41,45,46)(43,48,44,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19)(18,24)(20,21)(22,25)(23,31)(26,27)(28,29)(30,32)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(3,12)(7,8)(13,16)(14,15)(17,20,23,32)(18,25,28,26)(19,21,31,30)(22,29,27,24)(33,34,39,38)(35,40,36,42)(37,41,45,46)(43,47,44,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20)(18,26)(19,30)(21,31)(22,24)(23,32)(25,28)(27,29)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,21,23,30)(18,27,28,22)(19,20,31,32)(24,26,29,25)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,22,23,27)(18,21,28,30)(19,25,31,26)(20,29,32,24)(33,38,39,34)(35,42,36,40)(37,41,45,46)(43,47,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=h,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, 8653802);