Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20);(2,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)Polycyclic presentation:
2^9|a^2=f,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239534, #72240655, D4×#3025, … |
| maximal normal subgroup of | #72239534, #72240655, D4×#3025, … |
| maximal quotient of | #72239358, C2×#6437994, #72268293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345190);