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 | |
3954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,11)(6,9,15,16)(10,13,12,14)(17,18)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(35,43)(36,45)(39,46)(40,48);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(33,34,38,42)(35,39,43,46)(36,40,45,48)(37,41,44,47);(2,8)(3,4)(9,16)(10,12)(18,25)(20,24)(26,31)(27,30)(33,35,38,43)(34,39,42,46)(36,44,45,37)(40,47,48,41);(1,4,7,3)(2,5,8,11)(6,10,15,12)(9,14,16,13)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,36)(34,40)(35,44)(37,43)(38,45)(39,47)(41,46)(42,48);(1,6,7,15)(2,9,8,16)(3,10,4,12)(5,13,11,14)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)(33,37,38,44)(34,41,42,47)(35,36,43,45)(39,40,46,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=ghi,d^2=gi,f^2=gh,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, 8652957);