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 | #2812 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(3,11)(4,13)(8,14)(9,16)(18,22)(23,25)(24,29)(30,31)(34,40)(35,42)(41,44)(43,48);(2,7)(3,11)(9,16)(10,12)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000692);