Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C22 | C23×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,20)(12,17)(13,16)(14,27)(15,21)(18,26)(19,25)(22,33)(23,30)(24,31)(28,36)(29,40)(32,39)(34,38)(35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,28)(17,20)(18,29)(19,30)(24,36)(25,37)(26,38)(27,35)(31,33)(32,34)(39,40);(10,16)(13,26)(14,25)(15,28)(17,31)(18,19)(20,33)(22,38)(23,37)(24,39)(27,32)(29,30)(34,35)(36,40);(2,5)(6,8)(11,21)(15,20)(22,36)(23,37)(28,33)(29,34)(30,35)(38,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^b=df,g^c=gh,h^c=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72234320, #72234321, #72234331, … |
| maximal normal subgroup of | #72234320, #72234321, #72234331, … |
| maximal quotient of | #72320231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531186);