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 | C22×D4 | C24 |
| 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:
(4,12)(8,15)(11,16)(13,14)(18,23)(20,24)(26,29)(28,31)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,2,6,10)(3,5,9,7)(4,8,12,15)(11,13,16,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,15)(12,14)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352505);