Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,21,29,19)(11,16,12,24,23,17,26,31)(13,18,28,35,27,33,39,34)(22,30,25,37,36,32,38,40);(2,5)(6,8)(9,11,14,12,15,23,29,26)(10,16,20,24,21,17,19,31)(13,22,28,25,27,36,39,38)(18,30,35,37,33,32,34,40);(9,12,14,23,15,26,29,11)(10,17,19,24,21,16,20,31)(13,25,28,36,27,38,39,22)(18,32,34,37,33,30,35,40);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,22)(12,25)(14,28)(15,27)(16,30)(17,32)(19,34)(20,35)(21,33)(23,36)(24,37)(26,38)(29,39)(31,40);(13,27)(18,33)(22,36)(25,38)(28,39)(30,32)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239527, #72239548, #72264817, … |
| maximal normal subgroup of | #72239527, #72239548, #72264817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279354);