Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(10,12)(13,14,17,18)(15,19,20,16);(5,6)(7,8)(14,18)(16,19);(2,4)(6,8);(9,10)(11,12)(13,15)(14,16)(17,20)(18,19);(1,2)(3,4)(5,6)(7,8)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72231755, #72232117, #72232136, … |
| maximal normal subgroup of | #72231755, #72232117, #72232136, … |
| maximal quotient of | #72251005, #72251018, C2×#4306358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213554);