Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483449 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(11,19,20,31)(12,22)(13,24)(15,25,26,36)(17,30)(18,29)(21,34,33,32)(23,35)(27,39,38,37)(28,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,27)(22,30)(23,28)(24,29)(31,36)(32,37)(33,38)(34,39)(35,40);(3,8)(5,7)(9,11,12,19,13,21,22,32,14,20,23,31,24,33,35,34)(10,15,17,25,18,27,30,37,16,26,28,36,29,38,40,39);(2,6)(5,7)(10,16)(15,26)(17,28)(18,29)(25,36)(27,38)(30,40)(37,39)Polycyclic presentation:
2^9|a^2=g,c^2=f,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265463, #72265470, #72265488, … |
| maximal normal subgroup of | #72265463, #72265470, #72265488, … |
| maximal quotient of | #72320519, C2×#496669, #72343722, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403865);