Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,8,13,10,14,22,27,11,15,9,23,21,28,24,34)(7,16,18,29,20,30,32,36,17,12,19,25,31,26,33,35);(5,7,9,19,10,20,24,33,11,17,8,18,21,31,22,32)(6,12,13,25,14,26,27,35,15,16,23,29,28,30,34,36);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239418, #72239421, #72239432, … |
| maximal normal subgroup of | #72239418, #72239421, #72239432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53592);