Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C24 |
| commutator, socle | C22 | C23×C16 |
| prime radical | C23 | C22×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,25,21,26,33,43,22,27,34,44,36,46,55,65,23,28,35,45,37,47,56,66,38,48,57,67,58,68,73,78)(19,24,30,40,31,41,52,62,32,42,53,63,54,64,72,77,29,39,49,59,50,60,69,74,51,61,70,75,71,76,79,80);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,29)(25,40)(26,41)(27,42)(28,39)(33,52)(34,53)(35,49)(36,54)(37,50)(38,51)(43,62)(44,63)(45,59)(46,64)(47,60)(48,61)(55,72)(56,69)(57,70)(58,71)(65,77)(66,74)(67,75)(68,76)(73,79)(78,80);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(19,29)(24,39)(30,49)(31,50)(32,51)(40,59)(41,60)(42,61)(52,69)(53,70)(54,71)(62,74)(63,75)(64,76)(72,79)(77,80)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,d^a=de,c^b=ce,d^b=di176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420005);