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, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,12)(5,8,15,16)(10,11,13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(34,39)(35,36)(37,45)(47,48);(2,7)(3,9)(8,16)(10,13)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34,38,39)(35,41,36,40)(37,42,45,46)(43,48,44,47);(2,7)(3,9)(8,16)(10,13)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,35,38,36)(34,40,39,41)(37,43,45,44)(42,47,46,48);(1,3)(2,4)(5,13)(6,9)(7,12)(8,14)(10,15)(11,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,4)(2,3)(5,14)(6,12)(7,9)(8,13)(10,16)(11,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,5,6,15)(2,8,7,16)(3,10,9,13)(4,11,12,14)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=ghi,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, 8658773);