Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,8,9,7)(6,10,11,12);(2,4);(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,h^2=i,b^a=bf,c^a=cg,d^b=df,e^b=ei140 conjugacy classes — view the full character table.
| maximal subgroup of | #72310629, #72310640, #72310745, … |
| maximal normal subgroup of | #72310629, #72310640, #72310745, … |
| maximal quotient of | #72321997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287647);