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 | #2682 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,13,4)(5,6,9,15)(11,12,16,14)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34,39,40)(35,37,42,38)(36,44,47,41)(43,46,48,45);(1,3,7,13)(2,8,10,4)(5,11,9,16)(6,12,15,14)(20,28)(24,31)(27,32)(29,30)(34,40)(35,42)(41,44)(43,48);(1,4,7,8)(2,3,10,13)(5,14,9,12)(6,11,15,16)(17,19,22,25)(18,23,26,21)(20,27,28,32)(24,30,31,29)(33,35)(34,37)(36,43)(38,40)(39,42)(41,45)(44,46)(47,48);(1,5)(2,6)(3,11)(4,14)(7,9)(8,12)(10,15)(13,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,4)(2,3)(5,14)(6,11)(7,8)(9,12)(10,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37,39,38)(34,42,40,35)(36,45,47,46)(41,48,44,43);(1,6)(2,9)(3,12)(4,11)(5,10)(7,15)(8,16)(13,14)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38,39,37)(34,35,40,42)(36,46,47,45)(41,43,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440680);