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 | |
3824 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(20,28)(24,29)(27,32)(30,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(2,7)(4,14)(8,10)(11,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(36,44)(39,46)(42,48)(45,47);(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3,6,13)(2,8,7,10)(4,11,14,16)(5,15,9,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,15)(13,16)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,16,10,11)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412935);