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 | |
3870 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)(3,8,4,5)(6,9,14,16)(11,15,12,13)(18,19)(20,27)(21,29)(30,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,48)(45,46);(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(1,4,7,3)(2,8,10,5)(6,11,14,12)(9,13,16,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36,39,34)(35,42,37,40)(38,41,46,44)(43,47,45,48);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352180);