Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C2×D4⋊C8 |
| prime radical | #3050 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10,12,18,14,19,13,16)(7,15)(11,20);(5,6,11,10,17,13,7,16,9,14,20,19,8,12,15,18);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287284, #72319447, D4×#1336, … |
| maximal normal subgroup of | #72287284, #72319447, D4×#1336, … |
| maximal quotient of | #72287158, #72323153 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43796);